# Weierstrass Fractal Drums -I – A Glimpse of Complex Dimensions

Claire David, Michel Lapidus

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Claire David, Michel Lapidus. Weierstrass Fractal Drums -I – A Glimpse of Complex Dimensions. 2023. hal-03642326v2

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# Weierstrass Fractal Drums -I – A Glimpse of Complex Dimensions

Claire David, Michel Lapidus

<figure><img src="image_1.png" /></figure>

► To cite this version:

Claire David, Michel Lapidus. Weierstrass Fractal Drums -I – A Glimpse of Complex Dimensions. Advances in Mathematics, 2025. hal-03642326v5

HAL Id: hal-03642326

https://hal.sorbonne-universite.fr/hal-03642326v5

Submitted on 12 Sep 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Weierstrass Fractal Drums - I

## A Glimpse of Complex Dimensions

Claire David¹ and Michel L. Lapidus²*

June 12, 2023

¹ Sorbonne Université

CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005, Paris, France

² University of California, Riverside, Department of Mathematics,
Skye Hall, 900 University Ave., Riverside, CA 92521-0135, USA

### Abstract

We establish fractal tube formulae for the sequence of prefractal graphs which converge to the Weierstrass Curve, called *Weierstrass Iterated Fractal Drums* (in short, Weierstrass IFDs), and which give, for a suitable (and geometrically meaningful) sequence of values of the parameter $\epsilon$ tending to zero, explicit expressions for the volume of the associated $\epsilon$-neighborhoods. For this purpose, we prove new geometric properties of the Curve and of the associated function, in relation with its local Hölder and reverse Hölder continuity, with explicit estimates that had not been obtained before. We also show that the Codimension $2 - D_W$ is the optimal Hölder exponent for the Weierstrass function $W$, from which it follows that, as is well known, $W$ is nowhere differentiable. Then, the formula, that yields the expression of the $\epsilon$-neighborhood, consists of a fractal power series in $\epsilon$, with underlying exponents the Complex Codimensions of the sequence of prefractal graphs. This enables us to obtain the associated (local and global, effective) tube and distance fractal zeta functions, whose poles yield the corresponding set of Complex Dimensions. We prove that the Complex Dimensions – apart from 0 and $-2$ – are periodically distributed along countably many vertical lines, with the same oscillatory period. By considering the lower and upper (effective) Minkowski contents of the $m^{th}$ prefractal approximation to the Weierstrass Curve, which we prove to be strictly positive, we then show that the Weierstrass IFD is Minkowski nondegenerate, as well as not Minkowski measurable, but admits a nontrivial average Minkowski content – and that, as expected, the Minkowski dimension (or box dimension) $D_W$ is the Complex Dimension with maximal real part, and zero imaginary part. An interesting (and likely general) new phenomenon arising in our investigation is that, for all sufficiently large positive integers $m$, the Complex Dimensions of the $m^{th}$ prefractal approximation to the Weierstrass Curve are the same and coincide with the Complex Dimensions of the Weierstrass IFD.

MSC Classification: 11M41, 28A12, 28A75, 28A80.

<footnote>*The research of M. L. L. was supported by the Burton Jones Endowed Chair in Pure Mathematics, as well as by grants from the U. S. National Science Foundation.</footnote>
# Weierstrass fractal drums – I –
A glimpse of Complex Dimensions

<figure><img src="image_3.png" /></figure>

Claire David<sup>a,*,1</sup>, Michel L. Lapidus<sup>b,2</sup>

## ARTICLE INFO

**Article history:**
Received 11 July 2023
Received in revised form 31 August 2025
Accepted 3 September 2025
Available online xxxx
Communicated by Lars Olsen

MSC:

11M41

28A12

28A75

28A80

**Keywords:**

Weierstrass Curve

Prefractal approximations

Best Hölder exponent

Complex Dimensions of an Iterated

Fractal Drum IFD

Fractal tube formula

Effective local and global tube zeta function

## ABSTRACT

We establish *fractal tube formulas* for the sequence of prefractal graphs which converge to the Weierstrass Curve, called *Weierstrass Iterated Fractal Drums* (in short, Weierstrass IFDs), and which give, for a suitable (and geometrically meaningful) sequence of values of the parameter $\epsilon$ tending to zero, explicit expressions for the volume of the associated $\epsilon$-neighborhoods. For this purpose, we prove new geometric properties of the Curve and of the associated function, in relation with its local Hölder and reverse Hölder continuity, with explicit estimates that had not been obtained before. We also show that the Codimension $2 - D_W$ is the optimal Hölder exponent for the Weierstrass function $W$, from which it follows that, as is well known, $W$ is nowhere differentiable. Then, the formula, that yields the expression of the volume of the $\epsilon$-neighborhood, consists of a fractal power series in $\epsilon$, with underlying exponents the Complex Codimensions of the sequence of prefractal graphs. This enables us to obtain the associated (effective local and global) *tube fractal zeta functions*, whose poles yield the corresponding set of Complex Dimensions. We prove that the Complex Dimensions are exact

<footnote><sup>a</sup> Sorbonne Université, Université Paris Cité, CNRS, INRIA, Laboratoire Jacques-Louis Lions, LJLL, F-75005 Paris, France</footnote>

<footnote><sup>b</sup> University of California, Riverside, Department of Mathematics, Skye Hall, 900 University Ave., Riverside, CA 92521-0135, USA</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: Claire.David@sorbonne-universite.fr (C. David).</footnote>

<footnote n="1">¹ The research of C. D. and M. L. L. is supported by the MITI CNRS 80 PRIME, Conditions Extrêmes and by the ANR FRACTALS (ANR-24-CE45-3362).</footnote>

<footnote n="2">² The research of M. L. L. was supported by the Burton Jones Endowed Chair in Pure Mathematics, as well as by grants DMS-0707524 and DMS-1107750 from the U.S. National Science Foundation.</footnote>
**Keywords:** Weierstrass Curve, prefractal approximations, best Hölder exponent, iterated fractal drum (IFD), Complex Dimensions of an IFD, box-counting (or Minkowski) dimension, fractal tube formula, effective local and global tube zeta function, effective local and global distance zeta function, (upper, lower and average) Minkowski contents, Minkowski non-measurability, Minkowski nondegeneracy, nowhere differentiability.

# Contents

<table><tr><td>1 Introduction</td><td>2</td></tr><tr><td>2 Geometric Framework</td><td>7</td></tr><tr><td>3 Iterated Fractal Drums and Tubular Neighborhoods</td><td>35</td></tr><tr><td>4 Prefractal Tube Formulas, Complex Dimensions and Average Minkowski Content</td><td>61</td></tr><tr><td>4.1 Preliminaries</td><td>61</td></tr><tr><td>4.2 Prefractal Tube Formulas and Prefractal Effective Zeta Functions</td><td>65</td></tr><tr><td>4.3 Complex Dimensions</td><td>75</td></tr><tr><td>4.3.1 Main Results</td><td>75</td></tr><tr><td>4.3.2 Exceptional Cases</td><td>83</td></tr><tr><td>4.3.3 Possible Interpretation</td><td>85</td></tr><tr><td>4.3.4 Analogy with the General Theory of Complex Dimensions</td><td>87</td></tr><tr><td>4.4 Minkowski Dimension, Minkowski Nondegeneracy, and Average Minkowski Content</td><td>88</td></tr><tr><td>4.5 The Noninteger Case</td><td>94</td></tr><tr><td>5 Concluding Comments</td><td>95</td></tr></table>

# 1 Introduction

Among the so-called “pathological objects” that appeared in the XIXth century, the Weierstrass Curve (W-Curve) stands as one of the most fascinating and intriguing ones. At first, it was simply designed and thought of in order to be continuous everywhere, while being nowhere differentiable. Given $\lambda \in ]0, 1[$, and $b$ such that $\lambda b > 1 + \frac{3\pi}{2}$, the associated function is defined as the sum of the uniformly convergent trigonometric series

$$x \in \mathbb{R} \mapsto \sum_{n=0}^{\infty} \lambda^n \cos(\pi b^n x).$$

The original proof, by K. Weierstrass [Wei75], in the case where $b$ is an odd positive integer, can also be found in [Tit39] (pages 351-353). It has been completed by the one, now classical, given by G. H. Hardy [Har16], in the more general case, where $b$ is any real number such that $\lambda b > 1$.

As is discussed in [Dav22], the introduction of this function challenged all the existing theories that went back to André-Marie Ampère, and has led to the emergence of many new functions possessing the same type of properties.

History then left it aside for a while, before new discovered properties brought it back once again to the forefront. It happened, in particular, that, in addition to its nowhere differentiability, the function – and the associated Curve – have self-similarity properties. After the works of A. S. Besicovitch and H. D. Ursell [BU37], Benoît Mandelbrot [Man77], [Man83], particularly highlighted the fractal
and simple, as well as periodically distributed along countably many vertical lines, with the same oscillatory period.

© 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

## Contents

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>2. Geometric framework</td><td>9</td></tr><tr><td>3. Iterated fractal drums and tubular neighborhoods</td><td>27</td></tr><tr><td>3.1. The tubular neighborhoods, and associated geometric characteristic numbers</td><td>30</td></tr><tr><td>4. Complex dimensions of the Weierstrass IFD</td><td>48</td></tr><tr><td>4.1. Prefractal tube formulas and prefractal effective zeta functions</td><td>48</td></tr><tr><td>4.2. Complex dimensions</td><td>52</td></tr><tr><td>4.2.1. Possible interpretation</td><td>57</td></tr><tr><td>4.2.2. Analogy with the general theory of complex dimensions</td><td>58</td></tr><tr><td>4.3. The noninteger case</td><td>59</td></tr><tr><td>5. Concluding comments</td><td>60</td></tr><tr><td>Acknowledgments</td><td>62</td></tr><tr><td>Appendix A. Technical appendix</td><td>62</td></tr><tr><td>A.1. Proof of Property 2.10, stated on page 20</td><td>62</td></tr><tr><td>A.2. Proof of Property 2.11, stated on page 21</td><td>63</td></tr><tr><td>A.3. Proof of Corollary 2.17, stated on page 26</td><td>67</td></tr><tr><td>A.4. Proof of Property 2.19, stated on page 27</td><td>68</td></tr><tr><td>A.5. Proof of Corollary 3.3, stated on page 36</td><td>73</td></tr><tr><td>A.6. Proof of Theorem 4.3, stated on page 54</td><td>74</td></tr><tr><td>References</td><td>75</td></tr></table>

# 1. Introduction

Among the so-called “pathological objects” that appeared in the XIXth century, the Weierstrass Curve (W-Curve) stands as one of the most fascinating and intriguing ones. At first, it was simply designed and thought of in order to be continuous everywhere, while being nowhere differentiable. Given $\lambda \in ]0, 1[$, and $b$ such that $\lambda b > 1 + \frac{3\pi}{2}$, the associated function is defined as the sum of the uniformly convergent trigonometric series

$$x \in \mathbb{R} \mapsto \sum_{n=0}^{\infty} \lambda^n \cos(\pi b^n x).$$

The original proof, by K. Weierstrass [55], in the case where $b$ is an odd positive integer, can also be found in [54] (pages 351-353). It has been completed by the one, now classical, given by G. H. Hardy [17], in the more general case, where $b$ is any real number such that $\lambda b > 1$.

As is discussed in [6], the introduction of this function challenged all the existing theories that went back to André-Marie Ampère, and has led to the emergence of many new functions possessing the same type of properties.
properties of the Weierstrass Curve. He also conjectured that the Hausdorff dimension of the graph
is given by $D_W = 2 + \frac{\ln \lambda}{\ln b} = 2 - \ln_b \frac{1}{\lambda}$.

Interesting discussions and results in relation to this question may be found in the book of K. Fal-
coner [Fal86]. As for the box dimension, a first series of results have been obtained by J.-L. Kaplan,
J. Mallet-Paret and J. A. Yorke [KMPY84], where the authors show that it is equal to the Lyapunov
dimension of the equivalent attracting torus. Then, the problem was tackled by F. Przytycki and
M. Urbański [PU89], as well as by T.-Y. Hu and K.-S. Lau [HL93].

As for the Hausdorff dimension, the first key result was obtained by F. Ledrappier [Led92], where
the Curve is considered as “the repeller for some expanding self-mapping on $[0, 1] \times \mathbb{R}$”, in the case
where *b* is an integer, an assumption that is of importance, in so far as a Markov partition for the map-
ping *x* $\mapsto$ *bx* mod 1 is involved. The resulting dynamics thus obeys the Markov property, a fact that
has naturally led the author of [Led92] to using such notions as topological – metric entropies, explored
in his earlier joint work with L. S. Young [LY85]. An interesting and useful connection was therefore
established between Lyapunov exponents and dimensions, in this context. Another result was then
obtained by B. Hunt [Hun98] in 1998 in the case where arbitrary phases are included in each cosinu-
soidal term of the summation. Later, in 2014, K. Barański, B. Bárány and J. Romanowska [BBR14]
showed that, for any value of the real number *b*, there is a threshold value $\lambda_b$ belonging to the in-
terval $[\frac{1}{b}, 1[$ such that the Hausdorff dimension is equal to $D_W$, for every *b* in $]\lambda_b, 1[$. The results
obtained by W. Shen in [She18] went further than the main result of [BBR14] and, in fact, showed
that the Hausdorff dimension of the Weierstrass Curve is equal to $D_W$, for any (allowed) values of the
parameters. Furthermore, in [Kel17], G. Keller proposed a very original and much simpler proof of
the main results of [BBR14].

In [Dav18], the first author proved – in the case when $b = N_b$ is an integer, and in contrast to the
then existing work – that the Minkowski dimension (or box-counting dimension) of the Weierstrass
Curve could be obtained in a simple way, without requiring any theoretical background in dynamical
systems theory. The proof relies on the use of prefractal approximations; that is, here, a suitable se-
quence of finite graphs which converges towards the Weierstrass Curve. They are obtained by means
of a suitable nonlinear iterated function system (IFS) [Dav19], where, as in the case of the horse-
shoe attractor introduced by Stephen Smale, the nonlinear maps involved are not contractions, but
possess what can be viewed as an equivalent property, since, at each step of the iterative process,
they reduce the values of the two-dimensional Lebesgue measures of a given sequence of rectangles
covering the Curve. As expected, the Weierstrass Curve is invariant with respect to the family of those
maps, which provides us in this context with a result equivalent to the one that can be found in [BD85].

Interestingly, the intrinsic properties of the intriguing maps which constitute the nonlinear IFS
can be directly linked to the computation of the box dimension of the Weierstrass Curve, and to a
new proof of the nowhere differentiability of the Weierstrass function, as shown in [Dav22].

Yet, thus far, no connection has been established with the theory of Complex Dimensions. There-
fore, the following questions arise naturally in this setting: Can one prove that the Minkowski (or
box) dimension of the Weierstrass Curve is, also, a Complex Dimension? Can we also determine all of
the (possible) Complex Dimensions of this Curve, as well as obtain an associated fractal tube formula,
in the form of a fractal power series involving the underlying Complex Dimensions? (See [LRŽ17b],
Problem 6.2.24, page 560.)

The foundations of the theory of **Complex Dimensions** were laid by M. L. Lapidus and his col-
laborators in [Lap91], [Lap92], [Lap93], [LP93], [LM95], [LvF00], [LP06], [Lap08], [LPW11], [ELMR15],
[LvF06], [LRŽ17a], [LRŽ18], [Lap19], [HL21] and [Lap22], in particular. The theory provides a very
History then left it aside for a while, before new discovered properties brought it back once again to the forefront. It happened, in particular, that, in addition to its nowhere differentiability, the function – and the associated Curve – have self-similarity properties. After the works of A. S. Besicovitch and H. D. Ursell [3], Benoît Mandelbrot [46], [47], particularly highlighted the fractal properties of the Weierstrass Curve. He also conjec- tured that the Hausdorff dimension of the graph is given by $D_W = 2 + \frac{\ln \lambda}{\ln b} = 2 - \ln_b \frac{1}{\lambda}$, where $N_b = b \in \mathbb{N}^* = \mathbb{N} \setminus \{0\}$.

We point out that in his conjecture, he only used the phrase “fractal dimension”, which could also be interpreted as referring to the Minkowski (or box) dimension, for example.

Interesting discussions and results in relation to this question may be found in the book of K. Falconer [16]. As for the box dimension, a first series of results have been obtained by J.-L. Kaplan, J. Mallet-Paret and J. A. Yorke [24], where the authors show that it is equal to the Lyapunov dimension of the equivalent attracting torus. Then, the problem was tackled by F. Przytycki and M. Urbański [52], as well as by T.-Y. Hu and K.-S. Lau [18].

As for the Hausdorff dimension, the first key result was obtained by F. Ledrap- pier [31], where the Curve is considered as “the repeller for some expanding self-mapping on $[0, 1] \times \mathbb{R}$”, in the case where $b$ is an integer, an assumption that is of importance, in so far as a Markov partition for the mapping $x \mapsto bx \bmod 1$ is involved. The resulting dynamics thus obeys the Markov property, a fact that has naturally led the author of [31] to use such notions as topological – metric entropies, explored in his earlier joint work with L. S. Young [45]. An interesting and useful connection was therefore established between Lyapunov exponents and dimensions, in this context. Another result was then obtained by B. Hunt [20] in 1998 in the case where arbitrary phases are included in each cosinusoidal term of the summation. Later, in 2014, K. Barański, B. Bárany and J. Romanowska [1] showed that, for any value of the real number $b$, there is a threshold value $\lambda_b$ belonging to the interval $\left[\frac{1}{b}, 1\right[$ such that the Hausdorff dimension is equal to $D_W$, for every $b$ in $]\lambda_b, 1[$. The results obtained by W. Shen in [53] went further than the main result of [1] and, in fact, showed that the Hausdorff dimension of the Weier- strass Curve is equal to $D_W$, for any (allowed) values of the parameters. Furthermore, in [22], G. Keller proposed a very original and much simpler proof of the main results of [1].

In [4], the first author suggested – in the case when $b = N_b$ is an integer, and in contrast to the then existing work – that the Minkowski dimension (or box-counting dimension) of the Weierstrass Curve could be obtained in a simple way, without requiring any theoretical background in dynamical systems theory. The proposed proof relies on the use of prefractal approximations; that is, here, a suitable sequence of finite graphs which converges towards the Weierstrass Curve. They are obtained by means of a suitable nonlinear iterated function system (IFS) [5], where, as in the case of the horseshoe attractor introduced by Stephen Smale, the nonlinear maps involved are not contractions,
natural and intuitive way to characterize fractal strings or drums, in relation with their intrinsic vi-
brational properties. Geometrically, in the latter case, this means studying the oscillations of a small
neighborhood of the boundary, i.e., of a tubular neighborhood, where points are located within an
epsilon distance from any edge. As is explained in [Lap19], a fractal may be viewed "as a musical
instrument tuned to play certain notes with frequencies (respectively, amplitudes) essentially equal
to the real parts (respectively, the imaginary parts) of the underlying complex dimensions". One can
also imagine a "geometric wave propagating through the fractal" [Lap19].

The one-dimensional theory of Complex Dimensions (i.e., that of fractal strings) was developed,
in particular, in the books by the second author and M. van Frankenhuisjen [LvF00], [LvF06], where
general explicit formulas and fractal tube formulas were obtained for fractal strings (see [LvF06],
Chapters 5 and 8). Later, in the book [LRŽ17b] – as well as in a series of accompagnying papers,
including [LRŽ17a], [LRŽ17c] and [LRŽ18] – the higher-dimensional theory of Complex Dimensions
was developed by the second author, G. Radunovic and D. Žubrinić, in the general case of bounded
subsets of Euclidean space $\mathbb{R}^N$ and of relative fractal drums of $\mathbb{R}^N$, with $N \ge 1$ being an arbitrary
integer. General fractal tube formulas were also obtained in this context and applicable to a large
variety of examples; see [LRŽ17b], Chapter 5, and [LRŽ18]. In short, Complex Dimensions are defined
as the poles of the meromorphic continuation of suitable geometric or fractal zeta functions, associated
with the fractal under study. A geometric object is then said to be *fractal* if it admits at least one
nonreal Complex Dimension, thereby giving rise to geometric oscillations via the corresponding fractal
tube formula. For example, in agreement with one's intuition, the Devil's Staircase (i.e., the graph of
the Cantor-Lebesgue function) is shown to be fractal, in this sense, whereas it is not fractal according
to Benoît B. Mandelbrot's definition in [Man83], because its topological and Hausdorff dimensions
coincide.

Under a mild assumption, the (upper) Minkowski dimension of the geometric object under study
is equal to the abscissa of convergence of the geometric, distance or tube, fractal zeta functions, and is
the only Complex Dimension located on the real axis and with maximal real part, therefore giving rise,
via the corresponding fractal tube formula, to geometric, spectral, or dynamical oscillations with the
largest amplitudes. We note that fractal tube formulas express the volume of (small) $\varepsilon$-neighborhoods
of the fractal as a fractal power series, with exponents the underlying Complex Codimensions.

Building on the work on multifractal zeta functions and Complex Dimensions of multifractals
strings developed in [LR09], [LLVR09], [ELMR15], along with the work on Complex Dimensions and
fractal tube formulas in [LvF00], [LvF06]. L. O. R. Olsen [Ols13a], [Ols13b], also obtained a suitable
multifractal analog of fractal tube formulas in this context.

A clear summary of the theory of Complex Dimensions for fractal strings can be found in [Ols01],
while a long survey of the theory of Complex Dimensions, both for fractal strings and in higher di-
mensions, is given in [Lap19].

A question which naturally arises in this context is that of differential operators on such struc-
tures. In the case of fractal strings, as an echo to noncommutative geometry, where *spectral triples*
are involved, a *geometric zeta function* provides the set of complex modes, while the dimensions stand
as its nonreal poles. The occurrence of the zeta function can be understood very intuitively, in so far
as it simply represents the trace of the differential operator at a complex order $s$. Thus, the poles are
nothing but the maximal orders of differentiation. Hence, dimensions.

The notion of *a fractal drum* extends that of *a fractal string* to higher-dimensional Euclidean
spaces, and involves an open subset with a fractal boundary. In the Euclidean plane, this boundary is
a curve. The word “drum” calls for vibrations: intuitively, one understands that they occur in a small
neighborhood of the boundary, a tubular neighborhood, the Lebesgue measure of which is associated
but possess what can be viewed as an equivalent property, since, at each step of the
iterative process, they reduce the values of the two-dimensional Lebesgue measures of
a given sequence of rectangles covering the Curve. As expected, the Weierstrass Curve
is invariant with respect to the family of those maps, which provides us in this context
with a result equivalent to the one that can be found in [2].

Interestingly, the intrinsic properties of the intriguing maps which constitute the
nonlinear IFS can be directly linked to the computation of the box dimension of the
Weierstrass Curve, and to a new proof of the nowhere differentiability of the Weierstrass
function, as was suggested in [6]. This proposed proof has since been made fully rigorous
in [14], which builds on [6], as well as relies and expands upon the ideas and techniques
introduced in the present paper.

Yet, thus far, no connection has been established with the theory of Complex Dimensions. Therefore, the following questions arise naturally in this setting: Can one prove that the Minkowski (or box) dimension of the Weierstrass Curve is, also, a Complex Dimension? Can we also determine all of the (possible, or, better, exact) Complex Dimensions of this Curve, as well as obtain an associated fractal tube formula, in the form of a fractal power series involving the underlying Complex Dimensions? (See [39], Problem 6.2.24, page 560.)

The foundations of the theory of **Complex Dimensions** were laid by M. L. Lapidus and his collaborators in [25], [26], [27], [34], [33], [42], [35], [28], [36], [15], [43], [38], [41], [29], [19] and [30], in particular. The theory provides a very natural and intuitive way to characterize *fractal strings* or *drums*, in relation with their intrinsic vibrational properties. Geometrically, in the latter case, this means studying the oscillations of a small neighborhood of the boundary, i.e., of a tubular neighborhood, where points are located within an epsilon distance from any edge. As is explained in [29], a fractal may be viewed "as a musical instrument tuned to play certain notes with frequencies (respectively, amplitudes) essentially equal to the real parts (respectively, the imaginary parts) of the underlying complex dimensions". One can also imagine a "*geometric wave propagating* through the fractal" [29].

The one-dimensional theory of Complex Dimensions (i.e., that of fractal strings) was
developed, in particular, in the books by the second author and M. van Frankenhuis-
jen [42], [43], where general explicit formulas and fractal tube formulas were obtained for
fractal strings (see [43], Chapters 5 and 8). Later, in the book [39] – as well as in a series
of accompanying papers, including [38], [40] and [41] – the higher-dimensional theory of
Complex Dimensions was developed by the second author, G. Radunovic and D. Žubrinić,
in the general case of bounded subsets of Euclidean space R<sup>N</sup> and of relative fractal
drums of R<sup>N</sup>, with N ≥ 1 being an arbitrary integer. General fractal tube formulas were
also obtained in this context and applicable to a large variety of examples; see [39],
Chapter 5, and [41]. In short, Complex Dimensions are defined as the poles of the mero-
morphic continuation of suitable geometric or fractal zeta functions, associated with the
fractal under study. A geometric object is then said to be *fractal* if it admits at least
one *nonreal Complex Dimension*, thereby giving rise to geometric oscillations via the
to a *tube zeta function* which, similarly, enables one to obtain the Complex Dimensions, which stand
as characteristic numbers that account for specific geometric properties of the fractal boundary, here,
the underlying curve.

For the Koch Snowflake Curve, a *fractal tube formula* was obtained by M. L. Lapidus and E. P. J. Pear-
se in [LP06]. As was pointed out in [LRŽ17b] (see Problem 6.2.24, page 560), the case of **the Weier-**
**strass Curve** remained a *difficult open problem*, which we propose to solve in this paper. It is directly
associated to our previous work [Dav18], in so far as precise estimates are required for the elementary
heights of the sequence of natural prefractal approximations tending towards the Curve. As is often
the case in such a situation, we significantly improve these estimates, which also enable us to obtain
the exact values of the local extrema, and to determine the optimal Hölder exponent of $\mathcal{W}$. Those
extrema – which form a dense subset of the Weierstrass Curve – directly depend on the choice of an
initial set of points, which happen to be here the fixed points of the nonlinear iterated function system
involved in the construction of the Curve; see [Dav19] for further details. Moreover, we introduce the *concept of self-shape similarity*, a more general one than the standard notion of *self-similarity*.

The first novelty of our approach is that we define the Complex Dimensions of the Weierstrass Curve as the set of the Complex Dimensions of the sequence of $m^{th}$ prefractal graphs which converge to the Curve – *Weierstrass Iterated Fractal Drum* (in short, Weierstrass IFD), or, equivalently in our context, of the sequence of $m^{th}$ prefractal approximations which converge to the Curve. More specifically, we show that the set of (possible) Complex Dimensions is independent of the positive integer *m* sufficiently large. For this IFD, our tubular neighborhoods are located on both sides of the involved prefractals, which seems natural, because vibrations may occur on either side of the underlying fractal drum. However, when it comes to computing the associated fractal tube zeta function, classical methods, as in [LP06] and [LPW11] (see also [LvF00], §10.3, and [LvF06], §12.4), cannot be directly applied, since our fractal tube formulas can only be obtained for a sequence of characteristic lengths – the *cohomology infinitesimals*. More precisely, we only dispose of discrete values (but geometrically natural) for the fractal tube formulas, instead of an explicit expression of the tube formula on an interval of the form $[0, \epsilon_0]$, where $\epsilon_0 > 0$ stands for a small parameter. The second novelty of our work is that we bypass this difficulty, by means of Riemann sums, insofar as, when one passes to the limit, we can obtain the desired integral formula, which gives the fractal tube zeta function. Once the fractal tube formula has been obtained, we deduce from it the explicit form of the local and global fractal (tube and distance) zeta functions, along with the Complex Dimensions of the IFD, which are the same at any step of the process, for all prefractal approximations sufficiently close to the Weierstrass Curve. Note that the later results obtained in [DL23b] corroborate and further justify our approach. Indeed, not only the Complex Dimensions of the IFD are the same as the Complex Dimensions of the fractal involved, as is proved in [DL23b], but, also, the determination of the Complex Dimensions of the IFD is a compulsory step in order to know the Complex Dimensions of the limiting object – in our case, the Weierstrass Curve. In the process, we introduce the new notions of *effective tubular neighborhood*, as well as of *effective local and global fractal zeta functions*.

The main results obtained in this paper, where we consider the case $b = N_b$ being an integer, can
be found in the following places:

i. In Corollary 2.12, on page 24, and Theorem 2.13, on page 26, along with Corollary 2.14, on page 27, where we prove the sharp local Hölder continuity, and a sharp discrete version of reverse Hölder continuity, with optimal Hölder exponent, for the Weierstrass function $\mathcal{W}$, equal to the (Minkowski) Codimension $2 - D_{\mathcal{W}} = \ln_{N_b} \frac{1}{\lambda}$. It follows, in particular, that $\mathcal{W}$ is nowhere differentiable – as is well known, although our method of proof is completely different from the usual ones.

ii. In Theorem 4.7, on page 72 and Theorem 4.11, on page 83, which yield, for specific (and geo-
corresponding fractal tube formula. For example, in agreement with one's intuition, the Devil's Staircase (i.e., the graph of the Cantor-Lebesgue function) is shown to be fractal, in this sense, whereas it is not fractal according to Benoît B. Mandelbrot's definition in [47], because its topological and Hausdorff dimensions coincide.

Under a mild assumption, the (upper) Minkowski dimension of the geometric object under study is equal to the abscissa of convergence of the geometric, distance or tube, fractal zeta functions, and is the only Complex Dimension located on the real axis and with maximal real part, therefore giving rise, via the corresponding fractal tube formula, to geometric, spectral, or dynamical oscillations with the largest amplitude. We note that fractal tube formulas express the volume of (small) $\varepsilon$-neighborhoods of the fractal as a fractal power series, with exponents the underlying Complex Codimensions.

Building on the work on multifractal zeta functions and Complex Dimensions of multifractals strings developed in [37], [32], [15], along with the work on Complex Dimensions and fractal tube formulas in [42], [43]. L. O. R. Olsen [50], [51], also obtained a suitable multifractal analog of fractal tube formulas in this context.

A clear summary of the theory of Complex Dimensions for fractal strings can be found in [49], while a long survey of the theory of Complex Dimensions, both for fractal strings and in higher dimensions, is given in [29].

A question which naturally arises in this context is that of differential operators on such structures. In the case of fractal strings, as an echo to noncommutative geometry, where spectral triples are involved, a geometric zeta function provides the set of complex modes, while the dimensions stand as its (typically) nonreal poles. The occurrence of the zeta function can be understood very intuitively, in so far as it simply represents the trace of the differential operator at a complex order $s$. Thus, the poles are nothing but the maximal orders of differentiation. Hence, dimensions.

The notion of a fractal drum extends that of a fractal string to higher-dimensional Euclidean spaces, and involves an open subset with a fractal boundary. In the Euclidean plane, this boundary is a curve. The word “drum” calls for vibrations: intuitively, one understands that they occur in a small neighborhood of the boundary, a tubular neighborhood, the Lebesgue measure of which is associated to a tube zeta function which, similarly, enables one to obtain the Complex Dimensions, which stand as characteristic numbers that account for specific geometric properties of the fractal boundary, here, the underlying curve.

For the Koch Curve, a fractal tube formula was obtained by M. L. Lapidus and E. P. J. Pearse in [35]. However, no fractal zeta functions were associated to this tube formula. As was pointed out in [39] (see Problem 6.2.24, page 560), the case of the Weierstrass Curve remained a difficult open problem, which we propose to solve in this paper. It is directly associated to our previous work [4], in so far as precise estimates are required for the elementary heights of the sequence of natural prefractal approximations tending towards the Curve. As is often the case in such a situation, we significantly improve these estimates, which also enable us to obtain the exact values of the local extrema, and to determine the optimal Hölder exponent of the Weierstrass function $W$.
metrically significant) values of the positive parameter $\epsilon$, the expression of the area of the $\epsilon$-neighborhood of each $m^{th}$ prefractal graph approximation, for all sufficiently large positive integers $m$ – a Weierstrass Fractal Tube Formula, which (apart from two terms associated with the Complex Dimensions 0 and $-2$) consists of an expansion of the form

$$ \sum_{\alpha \text{ real part of a Complex Dimension}} \epsilon^{2-\alpha} G_{\alpha} \left( \ln N_b \left( \frac{1}{\epsilon} \right) \right), \quad (\star) $$

where, for any real part $\alpha$ of a Complex Dimension, $G_{\alpha}$ denotes a continuous and one-periodic function. Furthermore, for $\alpha = \alpha_{max} = D_W$, the Minkowski dimension of the Curve – i.e., for $\alpha$ being equal to the maximal real part of the Complex Dimensions of the Weierstrass IFD – the periodic function $G_{\alpha_{max}}$ is nonconstant, as well as bounded away from zero and infinity. As is the case in the general theory of fractal tube formulas (see [LvF06], [LRŽ17b], Chapter 8 and Chapter 5, respectively), the resulting fractal power series has for exponents the Complex Codimensions of the Weierstrass Curve. Observe that each nonconstant periodic function in $(\star)$ gives rise to multiplicatively periodic (or log-periodic) oscillations in the scaling variable $\epsilon$.

iii. In Theorem 4.10, on page 81, where we exhibit the possible Complex Dimensions of the Weierstrass IFD, as the poles of the associated (local and global) Tube Zeta Functions, themselves obtained in Theorem 4.8, on page 76. Equivalently, in the light of [LRŽ17a], [LRŽ17b], since $D_W < 2$, the Complex Dimensions are also the poles of the associated distance zeta functions. In particular, we show that the Complex Dimensions (other than 0 and $-2$) are all simple and periodically distributed (with the same period $p = \frac{2\pi}{\ln N_b}$, the natural oscillatory period of the Weierstrass Curve) along countably many vertical lines, with abscissae $D_W - k(2 - D_W)$ and $1 - 2k$, where $k$ in $\mathbb{N} = \{0, 1, 2, \dots\}$ is arbitrary. In addition, $-2$ and 0 are also Complex Dimensions, and they are simple.

iv. In Theorem 4.12, on page 90 and Corollary 4.13, on page 92, where we prove the nondegeneracy of the Weierstrass IFD, in the Minkowski sense (see [LRŽ17b]), coming from the fact that, for all sufficiently large positive integers $m$, the upper and lower (effective) Minkowski contents of the $m^{th}$ prefractal polygonal approximation to the Curve are respectively positive and finite. As a result, the Minkowski dimension (or box-counting dimension) $D_W$ of the Weierstrass IFD exists; i.e., the lower and upper Minkowski dimensions of the IFD coincide. Also, since the periodic function $G_{D_W}$ is not constant, it follows that the Weierstrass IFD is not Minkowski measurable. Moreover, we show that the (effective) average Minkowski content of the Weierstrass IFD exists, is positive and finite, as well as coincides with the average value of the periodic function $G_{D_W}$.

v. As a corollary of Theorem 4.12 (page 90), the fact that the number $D_W$ is both the Minkowski Dimension and a Complex Dimension of the Weierstrass IFD; see Corollary 4.13, on page 92.

vi. The fractality of the Weierstrass IFD, in the sense of [LvF06], [LRŽ17b], [Lap19]; i.e., the existence of nonreal Complex Dimensions (with real part $D_W$) giving rise to geometric oscillations, in the Fractal Tube Formula obtained in this paper (Theorem 4.7, on page 72 and Theorem 4.11, on page 83), as described in ii. above. In fact, in the terminology of [LvF06] and [LRŽ17b], the Weierstrass IFD is fractal in countably many dimensions $d_k$, with $d_k \to -\infty$, as $k \to \infty$.

The Minkowski dimension (or box dimension) of the Weierstrass Curve, $D_W$, coincides with the maximum value of the real parts of the Complex Dimensions of the IFD. By considering the lower
Those extrema – which form a dense subset of the Weierstrass Curve – directly depend on the choice of an initial set of points, which happen to be here the fixed points of the nonlinear iterated function system involved in the construction of the Curve; see [5] for further details. Moreover, we introduce the concept of **self-shape similarity**, a more general one than the standard notion of **self-similarity**.

The first novelty of our approach is that we define the Complex Dimensions of the Weierstrass Curve as the set of the Complex Dimensions of the sequence of $m^{th}$ prefractal graphs which converge to the Curve – *Weierstrass Iterated Fractal Drum* (in short, Weierstrass IFD), or, equivalently in our context, of the sequence of $m^{th}$ prefractal approximations which converge to the Curve. In addition, we precisely determine the set of Complex Dimensions of the $m^{th}$ prefractal approximation and show that it is strictly increasing with $m$, for all positive integers $m$ sufficiently large. For this same IFD, our tubular neighborhoods are located on both sides of the involved prefractals, which seems natural, because vibrations may occur on either side of the underlying fractal drum. However, when it comes to computing the associated fractal tube zeta function, classical methods, as in [35] and [36] (see also [42], Section 10.3, and [43], Section 12.4), cannot be directly applied, since our fractal tube formulas can only be obtained for a sequence of characteristic lengths – the *cohomology infinitesimals*, whose name comes from their occurrence in Taylor-like expansions obtained in our recent work on fractal cohomology [12], [10] (along with [44], Chapters 5 and 8, [39], Chapter 5 and [41]); the notation $\varepsilon_m^m$ is in relation with the notation $\varepsilon$ used in the previous work of the second author and his collaborators (e.g., in [42], [43], [44], [35], [36], [39], [40], [41], [29]) for the variable involved in the Complex Dimensions series expansions. More precisely, we only dispose of discrete (but geometrically natural) values for the fractal tube formulas, instead of an explicit expression of the tube formula on an interval of the form $[0, \varepsilon_0]$, where $\varepsilon_0 > 0$ stands for a small parameter. This difficulty can be overcome insofar as the knowledge of the expression for the volume at this discrete value is simply the trace of the continuous volume function corresponding to an evolving tubular neighborhood.

Another novelty of our work is that we also shed new light on the theory of Complex Dimensions itself. Indeed, thus far, the integers involved in the imaginary parts of the Complex Dimensions of a fractal were arbitrary, at least in the so-called *lattice self-similar case*. We presently show why this should no longer be the case. Building on the fact that the imaginary parts of the Complex Dimensions aim at characterizing the (frequencies of the) oscillations of the fractal under study, which are, also, connected to the evolution in scales – in real life (fractal-shaped living forms), new details keep on appearing with characteristic spatial oscillations – another adaptation of the classical theory was required, by means of *cohomological vertex integers*, which enable us to *label* each vertex of the prefractal graph at a given step of the prefractal approximation; see also our recent works [12], [11], [14]. By introducing *cross-scales paths*, we can, in some sense, *track* the evolution of vertices starting from the initial step of the prefractal approximation.
Minkowski content, which we prove to be strictly positive, we show that $D_W$ is, as expected, a Complex Dimension of the IFD. In fact, it is natural to expect that this is also true for the Complex Dimensions themselves, which will be shown in [DL23b] to be the same for the Weierstrass IFD and for the Weierstrass Curve.

We also briefly discuss, in Subsection 4.5, on page 94, the noninteger case, i.e., when $b$ is any positive real number satisfying $\lambda b > 1$. This case will be studied in detail in a future work.

Now, the determination of those dimensions, as important as it may be, is not an end in itself. In fact, the Complex Dimensions directly echo the fractal cohomological properties of the Curve, which is the subject of our second paper, [DL22c]. The results of this paper and of [DL22c] are announced in the survey article [DL22a], where their main results are presented in a summarized form.

## 2 Geometric Framework

Henceforth, we place ourselves in the Euclidean plane, equipped with a direct orthonormal frame. The usual Cartesian coordinates are denoted by $(x, y)$. The horizontal and vertical axes will be respectively referred to as $(x'x)$ and $(y'y)$.

### Notation 1 (Set of all Natural Numbers and Intervals).

As in Bourbaki [Bou04] (Appendix E. 143), we denote by $\mathbb{N} = \{0, 1, 2, \dots\}$ the set of all natural numbers, and set $\mathbb{N}^* = \mathbb{N} \setminus \{0\}$.

Given $a, b$ with $-\infty \le a \le b \le \infty$, $]a, b[ = (a, b)$ denotes an open interval, while, for example, $]a, b] = (a, b]$ denotes a half-open, half-closed interval.

### Notation 2 (Wave Inequality Symbol (see [Tao06], Preface, page xiv)).

Given two positive numbers $a$ and $b$, we will use the notation $a \le b$, when there exists a strictly positive constant $C$ such that $a \le Cb$, which is equivalent to $a = O(b)$. Note that in our forthcoming context, we will often use $O(1)$ to denote terms which depend on $m \in \mathbb{N}$, but are bounded away from $0$ and $\infty$; more precisely, those terms will always satisfy bounds of the following form

$$0 < \text{Constant}_{inf} \le O(1) \le \text{Constant}_{sup} < \infty, \qquad (\mathcal{R} 1)$$

where $\text{Constant}_{inf}$ and $\text{Constant}_{sup}$ denote strictly positive and finite constants.

### Notation 3 (Weierstrass Parameters).

In the sequel, $\lambda$ and $N_b$ are two real numbers such that

$$< \lambda < 1, \quad N_b \in \mathbb{N}^* \quad \text{and} \quad \lambda N_b > 1 \cdot \quad (\spadesuit) \qquad (\mathcal{R} 2)$$

As explained in [Dav19], we deliberately made the choice to introduce the notation $N_b$ which replaces the initial $b$, in so far as, in Hardy's paper [Har16] (in contrast to Weierstrass's original
proximation. In this light, our approach differs from the usual one – we *a priori* introduce the imaginary parts of the expected Complex Dimensions.

Recall that, in the classical theory of Complex Dimensions [42]–[44], [39], a geometric object is said to be *fractal* if it admits at least one nonreal Complex Dimension (defined as a pole of the associated geometric or fractal zeta function), where the imaginary part of the complex dimension characterizes the local oscillations (changes of direction) of the object structure in the neighborhood of points as one zooms into the object, i.e., as a function of the scale of observation of the object. We are still working along these lines: in our present work, our cross-scales paths and associated sequences of cohomological vertex integers characterize fractality, in the sense that they provide a faithful reflection of the change of shape and the occurrence of new details when switching from a prefractal graph approximation to the next one.

By relying on those geometric characterizations, we can thus obtain fractal tube formulas, and deduce from them the explicit form of the local and global fractal (tube and distance) zeta functions, along with the Complex Dimensions of the IFD, which are progressively appearing at every step of the process, for all prefractal approximations sufficiently close to the Weierstrass Curve. Note that the later results obtained in [14] corroborate and further justify our approach. Indeed, not only the Complex Dimensions of the IFD are the same as the Complex Dimensions of the fractal involved, as is proved in [14], but, also, the determination of the Complex Dimensions of the IFD is a compulsory step in order to know the Complex Dimensions of the limiting object – in our case, the Weierstrass Curve. In the process, we introduce the new notions of *effective tubular neighborhood*, as well as of *effective local and global fractal zeta functions*.

The main results obtained in this paper, where we consider the case when $b = N_b$ is an integer, can be found in the following places:

i. In Corollary 2.13, on page 22, and Theorem 2.14, on page 23, along with Corollary 2.15, on page 25, where we prove the sharp local Hölder continuity, and a sharp discrete version of reverse Hölder continuity, with optimal Hölder exponent, for the Weierstrass function $W$, equal to the (Minkowski) Codimension $2 - D_W = \ln_{N_b} \frac{1}{\lambda}$. It follows, in particular, that $W$ is nowhere differentiable – as is well known, although our method of proof is completely different from the usual ones.

ii. In Theorem 4.1, on page 48, which yields, for specific (and geometrically significant) values of the positive parameter $\epsilon$, the expression of the area of the $\epsilon$-neighborhood of each $m^{th}$ prefractal graph approximation, and for all sufficiently large positive integers $m$, a Weierstrass Fractal Tube Formula, which consists of an expansion of the form

$$ \sum_{\omega \text{ Complex Dimension}} c_{\omega} \epsilon^{\omega}, \qquad (*) $$
article [Wei75]), b is any positive real number satisfying $\lambda b > 1$, whereas we deal here with the specific case of a natural integer, which accounts for the natural notation $N_b$; see, however, Section 4.5.

**Definition 2.1 (Weierstrass Function, Weierstrass Curve).**

We consider the *Weierstrass function* $W$, defined, for any real number $x$, by

$$W(x) = \sum_{n=0}^{\infty} \lambda^n \cos(2\pi N_b^n x) . \qquad (\mathcal{R} 3)$$

We call the associated graph the *Weierstrass Curve*.

Due to the one-periodicity of the $W$-function, from now on, and without loss of generality, we restrict our study to the interval $[0, 1[ = [0, 1)$.

**Notation 4 (Logarithm).**

Given $y > 0$, $\ln y$ denotes the natural logarithm of $y$, while, given $a > 0$, $a \neq 1$, $\ln_a y = \frac{\ln y}{\ln a}$ denotes the logarithm of $y$ in base $a$; so that, in particular, $\ln = \ln_a$.

**Notation 5.** For the parameters $\lambda$ and $N_b$ satisfying condition (♣) (see Notation 3, on page 7), we denote by

$$D_W = 2 + \frac{\ln \lambda}{\ln N_b} = 2 - \ln_{N_b} \frac{1}{\lambda} \in ]1, 2[ \qquad (\mathcal{R} 4)$$

the box-counting dimension (or Minkowski dimension) of the Weierstrass Curve $\Gamma_W$, which happens to be equal to its Hausdorff dimension [KMPY84], [BBR14], [She18], [Kel17]. As was mentioned earlier, our results in this paper will also provide a direct geometric proof of the fact that $D_W$, the Minkowski dimension (or box-counting dimension) of $\Gamma_W$, exists and takes the above value.

*Remark 2.1.* As can be found, for instance, in [Fal86], we recall that the *box-counting dimension* (or *box dimension*, in short), of $\Gamma_W$, is given by

$$D_W = - \lim_{\delta \to 0^+} \frac{\ln N_\delta(\Gamma_W)}{\ln \delta}, \quad (\diamond)$$

where $N_\delta(\Gamma_W)$ stands for any of the following quantities:

i. the smallest number of sets of diameter at most $\delta$ that cover $\Gamma_W$ on $[0, 1[$;

ii. the smallest number of closed balls of radius $\delta$ that cover $\Gamma_W$ on $[0, 1[$;

iii. the smallest number of cubes of side $\delta$ that cover $\Gamma_W$ on $[0, 1[$;

iv. the number of $\delta$-mesh cubes that intersect $\Gamma_W$ on $[0, 1[$;

v. the largest number of disjoint balls of radius $\delta$ with centers in $\Gamma_W$ on $[0, 1[$.
where, for any Complex Dimension $\omega$, $c_\omega$ denotes a nonzero complex coefficient, coinciding with the residue at $\omega$ of the $m^{th}$ effective prefractal tube zeta function.

iii. In Theorem 4.2 on page 52, and in Corollary 4.4, on page 55, where, for all positive integers $m$ sufficiently large, we calculate the $m^{th}$ effective prefractal tube zeta function (associated with the $m^{th}$ prefractal approximation), as well as determine its set of exact poles (i.e., the set of exact Complex Dimensions of the $m^{th}$ prefractal approximation) and show that it strictly increases with $m$.

iv. In Theorem 4.3, on page 54, where we show (by using some key results on local and global polyhedral zeta functions from [14]) the existence of the effective global tube zeta function (defined as the pointwise and locally uniform limit as $m \to \infty$ of the $m^{th}$ effective prefractal tube zeta functions), which is meromorphic on all of $\mathbb{C}$, as well as has the Minkowski dimension $D_W$ of the Weierstrass Curve as its abscissa of convergence.

v. In Theorem 4.5, on page 56, where we exhibit the exact Complex Dimensions of the Weierstrass IFD, as the poles of the associated prefractal (or local) and global tube zeta functions, themselves obtained in Theorem 4.2, on page 52, and in Theorem 4.3, on page 54. In particular, we show that the Complex Dimensions are all exact, simple and periodically distributed (with the same period $p = \frac{2\pi}{\ln N_b}$, the natural oscillatory period of the Weierstrass Curve) along countably many vertical lines, with abscissae $D_W - k(2 - D_W)$, where $k$ in $\mathbb{N} = \{0, 1, 2, \dots\}$ is arbitrary.

vi. The fractality of the Weierstrass IFD, in the sense of [43], [39], [29]; i.e., the existence of nonreal Complex Dimensions (with real part $D_W$) giving rise to geometric oscillations, in the fractal tube formula obtained in this paper (Theorem 4.1, on page 48), as described in ii. above. In fact, in the terminology of [43], [44], [39] and [29], the Weierstrass IFD is fractal in countably many dimensions $d_k = D_W - k(2 - D_W)$, with $d_k \to -\infty$, as $k \to \infty$.

The Minkowski dimension (or box dimension) of the Weierstrass Curve, $D_W$, coincides with the maximum value of the real parts of the Complex Dimensions of the IFD. In fact, it is natural to expect that this is also true for the Complex Dimensions themselves, which are shown in [14] to be the same for the Weierstrass IFD and for the Weierstrass Curve.

We also briefly discuss, in Section 4.3, on page 59, the noninteger case, i.e., when $b$ is any positive real number satisfying $\lambda b > 1$. This case will be studied in detail in a future work.

Now, the determination of those dimensions, as important as it may be, is not an end in itself. In fact, the Complex Dimensions directly echo the fractal cohomological properties of the Curve, which is the subject of our second paper, [12].

The results of this paper and of [12] are announced in the survey article [9], where their main results are presented in a summarized form.
Furthermore, for the Weierstrass Curve $\Gamma_W$, as, more generally, for any bounded subset of Euclidean space – the box-counting dimension coincides with the Minkowski dimension.

We stress that our results will imply that the Minkowski (or box-counting) dimension of the Weierstrass Curve exists; more specifically, the above limit exists and is equal to $D_W = 2 + \frac{\ln \lambda}{\ln N_b}$.

**Convention (The Weierstrass Curve as a Cyclic Curve).**

In the sequel, we identify the points $(0, W(0))$ and $(1, W(1)) = (1, W(0))$. This is justified by the fact that the Weierstrass function $W$ is 1-periodic, since $N_b$ is an integer.

*Remark 2.2.* The above convention makes sense, because the points $(0, W(0))$ and $(1, W(1))$ have the same vertical coordinate, in addition to the periodic properties of the $W$-function.

**Property 2.1.** (Symmetry with Respect to the Vertical Line $x = \frac{1}{2}$)

Since, for any $x \in [0, 1]$,

$$W(1-x) = \sum_{n=0}^{\infty} \lambda^n \cos(2\pi N_b^n - 2\pi N_b^n x) = W(x),$$

the Weierstrass Curve is symmetric with respect to the vertical straight line $x = \frac{1}{2}$.

**Proposition 2.2 (Nonlinear and Noncontractive Iterated Function System (IFS)).**

Following our previous work [Dav18], we approximate the restriction $\Gamma_W$ to $[0, 1[ \times \mathbb{R}$, of the Weierstrass Curve, by a sequence of graphs, built via an iterative process. For this purpose, we use the nonlinear iterated function system (IFS) of the family of $C^\infty$ maps from $\mathbb{R}^2$ to $\mathbb{R}^2$ denoted by

$$\mathcal{T}_W = \{T_0, \dots, T_{N_b-1}\},$$

where, for any integer $i$ belonging to $\{0, \dots, N_b - 1\}$ and any point $(x, y)$ of $\mathbb{R}^2$,

$$T_i(x, y) = \left( \frac{x + i}{N_b}, \lambda y + \cos \left( 2\pi \left( \frac{x + i}{N_b} \right) \right) \right).$$

*Remark 2.3.* As is explained in [Dav19], it happens that the maps $T_i$, with $i = 0, \dots, N_b - 1$, comprising the IFS $\mathcal{T}_W$ in the statement of Proposition 2.2, on page 9 just above – are not contractions, in the classical sense. As a result, the nonlinearity of the IFS, $\mathcal{T}_W = \{T_i\}_{i=0}^{N_b-1}$, does not enable one to resort to the probabilistic approach of M. F. Barnsley and S. Demko [BD85], or to the earlier work of J. E. Hutchinson [Hut81], which is applicable in the case of standard fractals such as the Sierpiński Gasket and the Koch Curve. Interestingly, even if they are not contractions, our maps possess what can be viewed as satisfying an equivalent property, since, at each step of the iterative process, they reduce the two-dimensional Lebesgue measures of a given sequence of rectangles covering the Curve.
## 2. Geometric framework

Henceforth, we place ourselves in the Euclidean plane, equipped with a direct orthonormal frame. The usual Cartesian coordinates are denoted by $(x, y)$. The horizontal and vertical axes will be respectively referred to as $(Ox)$ and $(Oy)$.

**Notation 1 (Intervals).** Given $a, b$ with $-\infty \le a \le b \le \infty$, $]a, b[ = (a, b)$ denotes an open interval, while, for example, $]a, b] = (a, b]$ denotes a half-open, half-closed interval.

**Notation 2 (Weierstrass parameters).** In the sequel, $\lambda$ and $N_b$ are two real numbers such that

$$0 < \lambda < 1, \quad N_b \in \mathbb{N}^* \quad \text{and} \quad \lambda N_b > 1. \qquad (1)$$

As explained in [5], we deliberately made the choice to introduce the notation $N_b$ which replaces the initial $b$, in so far as, in Hardy's paper [17] (in contrast to Weierstrass's original article [55]), $b$ is any positive real number satisfying $\lambda b > 1$, whereas we deal here with the specific case of a positive integer, which accounts for the natural notation $N_b$; see, however, Section 4.3.

**Definition 2.1 (Weierstrass function, Weierstrass Curve).** We consider the Weierstrass function $W$, defined, for any real number $x$, by

$$W(x) = \sum_{n=0}^{\infty} \lambda^n \cos(2\pi N_b^n x). \qquad (2)$$

We call the associated graph the *Weierstrass Curve*.

Due to the one-periodicity of the $W$-function (since $N_b$ is an integer), from now on, and without loss of generality, we restrict our study to the interval $[0, 1[ = [0, 1)$.

**Notation 3.** For the parameters $\lambda$ and $N_b$ satisfying condition (1) (see Notation 2, on page 9), we denote by

$$D_W = 2 + \frac{\ln \lambda}{\ln N_b} = 2 - \ln N_b \frac{1}{\lambda} \in ]1, 2[ \qquad (3)$$

the box-counting dimension (or Minkowski dimension) of the Weierstrass Curve $\Gamma_W$, which happens to be equal to its Hausdorff dimension [24], [1], [53], [22]. As was mentioned earlier, our methods and results in this paper and in [14] will also provide a direct geometric proof of the fact that $D_W$, the Minkowski dimension (or box-counting dimension) of $\Gamma_W$, exists and takes the above value; see [14], Section 6.
This is due to the fact that they correspond, in a sense, to the composition of a contraction of ratio $r_x$ in the horizontal direction, and a dilatation of factor $r_y$ in the vertical direction, with $r_x r_y < 1$. Such maps are considered, for example, in the book of Robert L. Devaney [Dev03], where they play a part in the first step of the horseshoe map process introduced by Stephen Smale.

**Property 2.3 (Attractor of the IFS).**

The Weierstrass Curve is the attractor of the IFS $\mathcal{T}_W$: $\Gamma_W = \bigcup_{i=0}^{N_b-1} T_i(\Gamma_W)$.

*Proof.* We refer to our works [Dav18], [Dav19].

<figure><img src="image_1.png" /></figure>

**Notation 6 (Fixed Points).**

For any integer $i$ belonging to $\{0, \dots, N_b - 1\}$, we denote by

$$P_i = (x_i, y_i) = \left( \frac{i}{N_b - 1}, \frac{1}{1 - \lambda} \cos \left( \frac{2\pi i}{N_b - 1} \right) \right)$$

the unique fixed point of the map $T_i$ (see [Dav19]).

**Definition 2.2 (Sets of Vertices, Prefractals).**

We denote by $V_0$ the ordered set (according to increasing abscissae), of the points

$$\{P_0, \dots, P_{N_b-1}\}.$$

The set of points $V_0$ – where, for any $i$ of $\{0, \dots, N_b - 2\}$, the point $P_i$ is linked to the point $P_{i+1}$ – constitutes an oriented finite graph, ordered according to increasing abscissa, which we will denote by $\Gamma_{W_0}$. Then, $V_0$ is called *the set of vertices* of the graph $\Gamma_{W_0}$.

For any natural integer $m$, i.e., $m \in \mathbb{N}$, we set $V_m = \bigcup_{i=0}^{N_b-1} T_i(V_{m-1})$.

The set of points $V_m$, where two consecutive points are linked, is an oriented finite graph, ordered according to increasing abscissa, which we will call the **m**$^{th}$–**order W**–**prefractal**. Then, $V_m$ is called *the set of vertices* of the prefractal $\Gamma_{W_m}$; see Figures 1, 2, 3 on pages 11, 12, and 13.

**Definition 2.3 (Adjacent Vertices, Edge Relation).**

For any natural integer $m$, the prefractal graph $\Gamma_{W_m}$ is equipped with an edge relation $\sim_m$, as follows: two vertices $X$ and $Y$ of $\Gamma_{W_m}$, i.e. two points belonging to $V_m$, will be said to be *adjacent* (i.e., neighboring or junction points) if and only if the line segment $[X, Y]$ is an edge of $\Gamma_{W_m}$; we then
**Remark 2.1.** The definition of the box-counting dimension (or box dimension, in short) can be found, for instance, in [16]; note that for the Weierstrass Curve $\Gamma_W$, as, more generally, for any bounded subset of Euclidean space – the box-counting dimension coincides with the Minkowski dimension.

**Convention** (The Weierstrass Curve as a cyclic curve). In the sequel, we identify the points $(0, W(0))$ and $(1, W(1)) = (1, W(0))$. This is justified by the fact that the Weierstrass function $W$ is 1-periodic, since $N_b$ is an integer.

**Property 2.1** (Symmetry with respect to the vertical line $x = \frac{1}{2}$). Since, for any $x \in [0, 1]$,

$$W(1-x) = \sum_{n=0}^{\infty} \lambda^n \cos(2\pi N_b^n - 2\pi N_b^n x) = W(x),$$

the Weierstrass Curve is symmetric with respect to the vertical straight line $x = \frac{1}{2}$; see Fig. 7.

**Proposition 2.2** (Nonlinear and noncontractive iterated function system (IFS)). Following our previous work [4], we approximate the restriction $\Gamma_W$ to $[0, 1[\times\mathbb{R}$, of the Weierstrass Curve, by a sequence of graphs, built via an iterative process. For this purpose, we use the nonlinear iterated function system (IFS) consisting of the family of $C^\infty$ maps from $\mathbb{R}^2$ to $\mathbb{R}^2$ denoted by

$$\mathcal{T}_W = \{T_0, \dots, T_{N_b-1}\},$$

where, for any integer $i$ belonging to $\{0, \dots, N_b - 1\}$ and any point $(x, y)$ of $\mathbb{R}^2$,

$$T_i(x, y) = \left( \frac{x+i}{N_b}, \lambda y + \cos \left( 2\pi \left( \frac{x+i}{N_b} \right) \right) \right).$$

**Remark 2.2.** As is explained in [5], it turns out that the maps $T_i$, with $i = 0, \dots, N_b - 1$, comprising the IFS $\mathcal{T}_W$ in the statement of Proposition 2.2, on page 10 just above – are not contractions, in the classical sense. As a result, the nonlinearity of the IFS, $\mathcal{T}_W = \{T_i\}_{i=0}^{N_b-1}$, does not enable one to resort to the probabilistic approach of M. F. Barnsley and S. Demko [2], or to the earlier work of J. E. Hutchinson [21], which is applicable in the case of standard fractals such as the Sierpiński Gasket and the Koch Curve. Interestingly, even if they are not contractions, our maps possess what can be viewed as satisfying an equivalent property, since, at each step of the iterative process, they reduce the two-dimensional Lebesgue measures of a given sequence of rectangles covering the Curve. This is due to the fact that they correspond, in a sense, to the composition of a contraction of ratio $r_x < 1$ in the horizontal direction, and an expanding map of factor $r_y > 1$ in the vertical direction, with $r_x r_y < 1$. Such maps are considered, for example, in the book of Robert L. Devaney [7], where they play a role in the first step of the horseshoe map process introduced by Stephen Smale.
write $X \sim Y$. Note that this edge relation depends on $m$, which means that points adjacent in $V_m$ might not remain adjacent in $V_{m+1}$.

<figure><img src="image_2.png" /><figcaption>Figure 1: The prefractal graphs $\Gamma_{W_0}, \Gamma_{W_1}, \Gamma_{W_2}, \Gamma_{W_3}, \Gamma_{W_4}, \Gamma_{W_5}$, in the case where $\lambda = \frac{1}{2}$, and $N_b = 3$. For example, $\Gamma_{W_1}$ is on the right side of the top row, while $\Gamma_{W_4}$ is on the left side of the bottom row.</figcaption></figure>
**Property 2.3 (Attractor of the IFS).** The Weierstrass Curve is the unique attractor of the IFS $\mathcal{T}_W$: $\Gamma_W = \bigcup_{i=0}^{N_b-1} T_i(\Gamma_W)$.

**Proof.** We refer to our works [4], [5]. □

**Definition 2.2 (Fixed points).** For any integer $i$ belonging to $\{0, \dots, N_b - 1\}$, we denote by

$$P_i = (x_i, y_i) = \left( \frac{i}{N_b - 1}, \frac{1}{1 - \lambda} \cos \left( \frac{2\pi i}{N_b - 1} \right) \right)$$

the unique fixed point of the map $T_i$ (see [5]).

**Definition 2.3 (Sets of vertices, prefraçals).** We denote by $V_0$ the ordered set (according to increasing abscissae), of the points

$$\{P_0, \dots, P_{N_b-1}\}.$$

The set of points $V_0$ – where, for any $i$ of $\{0, \dots, N_b - 2\}$, the point $P_i$ is linked to the point $P_{i+1}$ – constitutes an oriented finite graph, ordered according to increasing abscissae, which we will denote by $\Gamma_0$. Then, $V_0$ is called the set of vertices of the graph $\Gamma_0$.

For any positive integer $m$, i.e., for $m \in \mathbb{N}^*$, we set $V_m = \bigcup_{i=0}^{N_b-1} T_i(V_{m-1})$.

We denote by $\Gamma_m$ the graph of the piecewise linear function connecting the points of $V_m$, ordered according to increasing abscissae. We call $\Gamma_m$ the **m**$^{th}$ **order W**-prefractal (or $m^{th}$ prefractal graph approximation). Accordingly, $\Gamma_m$ can thus be identified with an oriented, finite graph having $V_m$ as its set of vertices, while the edges of this oriented graph (whose union is identified with $\Gamma_m$) are the oriented line segments connecting consecutive vertices in $\Gamma_m$. Hence, $V_m$ is called the set of vertices of the prefractal $\Gamma_m$; see Figs. 1, 2, 3 on pages 12, 13, and 14.

**Property 2.4.** Given $m \in \mathbb{N}$, the number of elements in the finite set $V_m$ is

$$\#V_m = (N_b - 1) N_b^m + 1.$$

**Property 2.5 (Density of the set $V^* = \bigcup_{n \in \mathbb{N}} V_n$ in the Weierstrass Curve [12]).** The set $V^* = \bigcup_{n \in \mathbb{N}} V_n$ is dense in the Weierstrass Curve $\Gamma_W$.

**Definition 2.4 (Adjacent Vertices, Edge Relation).** For any $m \in \mathbb{N}$, the prefractal graph $\Gamma_m$ is equipped with an edge relation $\sim_m$, as follows: two vertices X and Y of $\Gamma_m$, i.e. two points belonging to $V_m$, are said to be **adjacent** (i.e., neighboring or junction
<figure><img src="image_3.png" /><figcaption>Figure 2: The prefractal graphs $\Gamma_{W_0}, \Gamma_{W_1}, \Gamma_{W_2}, \Gamma_{W_3}, \Gamma_{W_4}, \Gamma_{W_5}$, in the case where $\lambda = \frac{1}{2}$ and $N_b = 4$.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 1. The prefractal graphs $\Gamma_0, \Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4, \Gamma_5$, in the case where $\lambda = \frac{1}{2}$, and $N_b = 3$. For example, $\Gamma_1$ is on the right side of the top row, while $\Gamma_4$ is on the left side of the bottom row.</figcaption></figure>

points) if and only if the line segment $[X, Y]$ is an edge of $\Gamma_m$; we then write $X \sim_m Y$. Note that this edge relation depends on $m$, which means that points adjacent in $V_m$ might not remain adjacent in $V_{m+1}$.

**Definition 2.5** (Vertices of the prefractals, elementary lengths, heights and angles). Given a strictly positive integer $m$, we denote by $(M_{j,m})_{0 \le j \le (N_b-1)} N_b^m - 1$ the set of vertices of the prefractal graph $\Gamma_m$. For any integer $j$ in $\{0, \dots, (N_b-1) N_b^m - 1\}$, we denote by $(x_{j,m}, W(x_{j,m}))$ the coordinates of the vertex $M_{j,m}$:
<figure><img src="image_4.png" /><figcaption>Figure 3: The prefractal graphs $\Gamma_{W_0}, \Gamma_{W_1}, \Gamma_{W_2}, \Gamma_{W_3}, \Gamma_{W_4}, \Gamma_{W_5}$, in the case where $\lambda = \frac{1}{2}$ and $N_b = 7$.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 2. The prefractal graphs $\Gamma_0, \Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4, \Gamma_5$, in the case where $\lambda = \frac{1}{2}$ and $N_b = 4$.</figcaption></figure>

$$M_{j,m} = (x_{j,m}, W(x_{j,m})) = \left( \frac{j}{(N_b - 1) N_b^m}, W \left( \frac{j}{(N_b - 1) N_b^m} \right) \right).$$

We also introduce, for any integer $j$ in $\{0, \dots, (N_b - 1) N_b^m - 2\}$, the following quantities:
**Property 2.4.** [Dav18]

For any $m \in \mathbb{N}$, the following statements hold:

i. $V_m \subset V_{m+1}$.

ii. $\#V_m = (N_b - 1) N_b^m + 1$, where $\#V_m$ denotes the number of elements in the finite set $V_m$.

iii. The prefractal graph $\Gamma_{W_m}$ has exactly $(N_b - 1) N_b^m$ edges.

iv. The consecutive vertices of the prefractal graph $\Gamma_{W_m}$ are the vertices of $N_b^m$ simple nonregular polygons $P_{m,k}$ with $N_b$ sides. For any strictly positive integer $m$, the junction point between two consecutive polygons is the point

$$ \left( \frac{(N_b - 1)k}{(N_b - 1)N_b^m}, W \left( \frac{(N_b - 1)k}{(N_b - 1)N_b^m} \right) \right), \quad 1 \le k \le N_b^m - 1. $$

Hence, the total number of junction points is $N_b^m - 1$. For instance, in the case $N_b = 3$, the polygons are all triangles; see Figure 4, on page 14.

In the sequel, we will denote by $P_0$ **the initial polygon**, whose vertices are the fixed points of the maps $T_i$, $0 \le i \le N_b - 1$, introduced in Definition 2.2, on page 10, i.e., $\{P_0, \dots, P_{N_b-1}\}$.

<figure><img src="image_5.png" /><figcaption>Figure 4: The initial polygon $P_0$, and the polygons $P_{0,1}$, $P_{1,1}$, $P_{1,2}$, in the case where $\lambda = \frac{1}{2}$ and $N_b = 3$.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 3. The prefractal graphs $\Gamma_0, \Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4, \Gamma_5$, in the case where $\lambda = \frac{1}{2}$ and $N_b = 7$.</figcaption></figure>

i. the elementary horizontal lengths:

$$L_m = \frac{1}{(N_b - 1) N_b^m};$$

ii. the elementary lengths:

$$l_{j,m} = d(M_{j,m}, M_{j+1,m}) = \sqrt{L_m^2 + h_{j,m}^2},$$
**Definition 2.4 (Vertices of the Prefractals, Elementary Lengths, Heights and Angles).**

Given a strictly positive integer $m$, we denote by $(M_{j,m})_{0 \le j \le (N_b-1)N_b^m-1}$ the set of vertices of the prefractal graph $\Gamma_{W_m}$. One thus has, for any integer $j$ in $\{0, \dots, (N_b-1)N_b^m-1\}$,

$$M_{j,m} = \left( \frac{j}{(N_b-1)N_b^m}, W\left( \frac{j}{(N_b-1)N_b^m} \right) \right).$$

We also introduce, for any integer $j$ in $\{0, \dots, (N_b-1)N_b^m-2\}$, the following quantities:

i. the elementary horizontal lengths:

$$L_m = \frac{j}{(N_b - 1) N_b^m};$$

ii. the elementary lengths:

$$\ell_{j,j+1,m} = d(M_{j,m}, M_{j+1,m}) = \sqrt{L_m^2 + h_{j,j+1,m}^2},$$

where $h_{j,j+1,m}$ is defined in iii. just below.

iii. the elementary heights:

$$h_{j,j+1,m} = \left| W\left( \frac{j+1}{(N_b-1)N_b^m} \right) - W\left( \frac{j}{(N_b-1)N_b^m} \right) \right|;$$

iv. the minimal height:

$$h_m^{\inf} = \inf_{0 \le j \le (N_b-1)N_b^m-1} h_{j,j+1,m}, \qquad (\mathcal{R} 5)$$

along with the maximal height:

$$h_m = \sup_{0 \le j \le (N_b-1)N_b^m-1} h_{j,j+1,m}, \qquad (\mathcal{R} 6)$$
where $h_{j,m}$ is defined in iii. just below.

iii. the elementary heights:

$$h_{j,m} = |\mathcal{W}(x_{j+1,m}) - \mathcal{W}(x_{j,m})|;$$

Note that the absolute value is required, depending on the value taken by $j$. See Property 2.9, on page 19, along with Corollary 2.12, on page 21, and Corollary 2.13, on page 22.

iv. the minimal height:

$$h_m^{\inf} = \inf_{0 \le j \le (N_b - 1) N_b^m - 1} h_{j,m}, \quad (4)$$

along with the maximal height:

$$h_m = \sup_{0 \le j \le (N_b - 1) N_b^m - 1} h_{j,m}; \quad (5)$$

v. the geometric angles:

$$\theta_{j,m} = ((Oy), (\widehat{M_{j,m}M_{j+1,m}})),$$

where $(Oy)$ denotes the vertical axis, which yield the following value of the geometric angle between consecutive edges, namely, $[M_{j-1,m} M_{j,m}, M_{j,m} M_{j+1,m}]$, with $\arctan = \tan^{-1}$:

$$\theta_{j-1,m} + \theta_{j,m} = \arctan \frac{L_m}{h_{j-1,m}} + \arctan \frac{L_m}{h_{j,m}}.$$

Note that, of course, $\theta_{j,m} = \arctan \frac{L_m}{h_{j,m}}$.

**Property 2.6.** For the geometric angle $\theta_{j,m}$, with $0 \le j \le (N_b - 1) N_b^m - 2$ and $m \in \mathbb{N}$, we have the following relation:

$$\tan \theta_{j,m} = \frac{L_m}{h_{j,m}}.$$

**Property 2.7.** [4] For any $m \in \mathbb{N}$, the following statements hold:

i. $V_m \subset V_{m+1}$.

ii. The prefractal graph $\Gamma_m$ has exactly $(N_b - 1) N_b^m$ edges.

iii. The consecutive vertices of the prefractal graph $\Gamma_m$ are the vertices of $N_b^m$ simple nonregular polygons $P_{m,k}$ with $N_b$ sides, where $1 \le k \le N_b^m - 1$. (Note that contrary to the numbering of the vertices of the prefractal graph $\Gamma_m$, the numbering of the
v. the geometric angles:

$$ \theta_{j,j+1,m} = ((y'y), (\overline{M_{j,m}M_{j+1,m}})), $$

which yield the value of the geometric angle between consecutive edges $[M_{j-1,m} M_{j,m} M_{j,m} M_{j+1,m}]$:

$$ \theta_{j-1,j,m} + \theta_{j,j+1,m} = \arctan \frac{L_m}{|h_{j-1,j,m}|} + \arctan \frac{L_m}{|h_{j,j+1,m}|}. $$

**Property 2.5.** For the geometric angle $\theta_{j-1,j,m}$, with $0 \le j \le (N_b - 1) N_b^m - 1$ and $m \in \mathbb{N}$, we have the following relation:

$$ \tan \theta_{j-1,j,m} = \frac{h_{j-1,j,m}}{L_m}. $$

One now requires, at a given step $m \in \mathbb{N}^*$, the exact coordinates of the vertices of the prefactal graph $\Gamma_{W_m}$, i.e. of the following set of points:

$$ \left( \frac{j}{(N_b - 1) N_b^m}, W\left( \frac{j}{(N_b - 1) N_b^m} \right) \right) , \quad 0 \le j \le \#V_m. $$

Thus far, they could not be found in the existing literature on the subject.

For this purpose, it is interesting to use the scaling properties of the Weierstrass function.

**Property 2.6 (Scaling Properties of the Weierstrass Function, and Consequences).**

Since, for any real number $x$, $W(x) = \sum_{n=0}^{\infty} \lambda^n \cos(2\pi N_b^n x)$, one also has

$$ W(N_b x) = \sum_{n=0}^{\infty} \lambda^n \cos(2\pi N_b^{n+1} x) = \frac{1}{\lambda} \sum_{n=1}^{\infty} \lambda^n \cos(2\pi N_b^n x) = \frac{1}{\lambda} (W(x) - \cos(2\pi x)), $$

which yields, for any strictly positive integer $m$ and any $j$ in $\{0, \dots, \#V_m\}$,

$$ W\left(\frac{j}{(N_b - 1) N_b^m}\right) = \lambda W\left(\frac{j}{(N_b - 1) N_b^{m-1}}\right) + \cos\left(\frac{2\pi j}{(N_b - 1) N_b^m}\right). $$

By induction, one then obtains that

$$ W\left(\frac{j}{(N_b - 1) N_b^m}\right) = \lambda^m W\left(\frac{j}{N_b - 1}\right) + \sum_{k=0}^{m-1} \lambda^k \cos\left(\frac{2\pi N_b^k j}{(N_b - 1) N_b^m}\right). $$
polygons starts with $k = 1$ and not with $k = 0$.) For $1 \le k \le N_b^m - 1$, the vertices of $P_{m,k}$ are the following points,

$$ (x_{(N_b-1)k+k',m}, W(x_{(N_b-1)k+k',m})) , \quad 0 \le k' \le N_b - 1. $$

Thus, for any strictly positive integer $m$, the junction point between two consecutive polygons $P_{m,k}$ and $P_{m,k+1}$ is the point

$$ (x_{(N_b-1)k,m}, W(x_{(N_b-1)k,m})) , \quad 1 \le k \le N_b^m - 1. $$

Hence, the total number of junction points is $N_b^m - 1$. For instance, in the case $N_b = 3$, the polygons are all triangles; see Fig. 4, on page 17.

We call extreme vertices of the polygon $P_{m,k}$ the junction points

$$ V_{initial}(P_{m,k}) = (x_{(N_b-1)k,m}, W(x_{(N_b-1)k,m})) , \quad 0 \le k \le N_b^m - 1, $$

and

$$ V_{end}(P_{m,k}) = (x_{(N_b-1)(k+1),m}, W(x_{(N_b-1)(k+1),m})) , \quad 0 \le k \le N_b^m - 2. $$

In the sequel, we will denote by $P_0$ **the initial polygon**, whose vertices are the fixed points of the maps $T_i$, $0 \le i \le N_b - 1$, introduced in Definition 2.2, on page 11, i.e., $\{P_0, \dots, P_{N_b-1}\}$; see, again, Fig. 4.

In the same way, the consecutive vertices of the prefractal graph $\Gamma_m$, distinct from the fixed points $P_0$ and $P_{N_b-1}$ (see Definition 2.2, on page 11), are also the vertices of $N_b^m - 1$ simple nonregular polygons $Q_{m,j}$, for $1 \le j \le N_b^m - 2$. For any integer $j$ such that $1 \le j \le N_b^m - 2$, one obtains each polygon $Q_{m,j}$ by connecting the point number $j$ (i.e., with the notation of Definition 2.5, on page 12, the vertex $M_{j,m}$) to the point number $j+1$ (i.e., the vertex $M_{j+1,m}$) if $j \equiv i \mod N_b$, for $1 \le i \le N_b - 1$, and the point number $j$ to the point number $j - N_b + 1$ if $j \equiv 0 \mod N_b$.

For $1 \le k \le N_b^m - 1$, the vertices of $Q_{m,k}$ are the following points:

a. When the integer $N_b$ is odd:

$$ (x_{(N_b-1)k+\frac{N_b-1}{2}+k',m}, W(x_{(N_b-1)k+\frac{N_b-1}{2}+k',m})) , \quad 0 \le k' \le N_b - 1. $$

b. When the integer $N_b$ is even:

$$ (x_{(N_b-1)k+\frac{N_b}{2}+k',m}, W(x_{(N_b-1)k+\frac{N_b}{2}+k',m})) , \quad 0 \le k' \le N_b - 2. $$

As previously, we call extreme vertices of the polygon $Q_{m,k}$ the points

$$ V_{initial}(Q_{m,k}) = (x_{(N_b-1)k,m}, W(x_{(N_b-1)k,m})) , \quad 1 \le k \le N_b^m - 1, $$
**Property 2.7 (A Consequence of the Symmetry with Respect to the Vertical Line $x = \frac{1}{2}$).**

For any strictly positive integer $m$ and any $j$ in $\{0, \dots, \#V_m\}$, we have that

$$W\left(\frac{j}{(N_b - 1) N_b^m}\right) = W\left(\frac{(N_b - 1) N_b^m - j}{(N_b - 1) N_b^m}\right),$$

which means that the points

$$\left(\frac{(N_b - 1) N_b^m - j}{(N_b - 1) N_b^m}, W\left(\frac{(N_b - 1) N_b^m - j}{(N_b - 1) N_b^m}\right)\right) \quad \text{and} \quad \left(\frac{j}{(N_b - 1) N_b^m}, W\left(\frac{j}{(N_b - 1) N_b^m}\right)\right)$$

are symmetric with respect to the vertical line $x = \frac{1}{2}$.

**Definition 2.5 (Left-Side and Right-Side Vertices).**

Given natural integers $m, k$ such that $0 \le k \le N_b^m - 1$, and a polygon $P_{m,k}$, we define:

i. The set of its *left-side vertices* as the set of the first $\lfloor \frac{N_b - 1}{2} \rfloor$ vertices, where $[y]$ denotes the integer part of the real number $y$.

ii. The set of its *right-side vertices* as the set of the last $\lfloor \frac{N_b - 1}{2} \rfloor$ vertices.

When the integer $N_b$ is odd, we define the bottom vertex as the $(\frac{N_b - 1}{2})^{th}$ one; see Figure 6, on page 18.

<figure><img src="image_6.png" /><figcaption>Figure 5: Symmetric points with respect to the vertical line $x = \frac{1}{2}$.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 4. The initial polygon $P_0$, and the respective polygons $P_{1,0}$, $P_{1,1}$, $P_{1,2}$, $Q_{1,1}$, $Q_{1,2}$, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$. (See also Fig. 1, on page 12.)</figcaption></figure>

and

$$ \mathcal{V}_{\text{end}}(Q_{m,k}) = (x_{(N_b-1)(k+1),m}, W(x_{(N_b-1)(k+1),m})) , \quad 1 \le k \le N_b^m - 2. $$

**Definition 2.6** (Polygonal sets, polyhedral neighborhoods [14]). For any $m \in \mathbb{N}$, we introduce the following polygonal sets

$$ \mathcal{P}_m = \bigcup_{k=0}^{N_b^m-1} \mathcal{P}_{m,k} \subset \mathbb{R}^2 \quad \text{and} \quad \mathcal{Q}_m = \bigcup_{k=0}^{N_b^m-2} \mathcal{Q}_{m,k} \subset \mathbb{R}^2. $$

We then introduce the sequence of sets delimited by the Weierstrass IFD, or polygonal neighborhood of the Weierstrass Curve as the sequence $(D(\Gamma_m))_{m \in \mathbb{N}}$ of closed polygonal sets $(\mathcal{P}_m \cup \mathcal{Q}_m)_{m \in \mathbb{N}}$, where, for each $m \in \mathbb{N}$, $\mathcal{P}_m$ and $\mathcal{Q}_m$ respectively denote the polygonal sets introduced just above. Note that, consequently, for $m \in \mathbb{N}$, $D(\Gamma_m)$ consists of the union of the family of polygons $\mathcal{P}_m \cup \mathcal{Q}_m$.
<figure><img src="image_7.png" /><figcaption>Figure 6: The left-side and right-side vertices.</figcaption></figure>

**Property 2.8.** Since, for any natural integer $n$,

$$N_b^n = (1 + N_b - 1)^n = \sum_{k=0}^{n} \binom{n}{k} (N_b - 1)^k \equiv 1 \quad \mod N_b - 1,$$

one obtains, for any integer $j$ in $\{0, \dots, N_b - 1\}$:

$$W\left(\frac{j}{N_b - 1}\right) = \sum_{n=0}^{\infty} \lambda^n \cos\left(2\pi N_b^n \frac{j}{N_b - 1}\right) = \sum_{n=0}^{\infty} \lambda^n \cos\left(\frac{2\pi j}{N_b - 1}\right) = \frac{1}{1 - \lambda} \cos\left(\frac{2\pi j}{N_b - 1}\right).$$

We observe that the point

$$\left(\frac{j}{N_b - 1}, W\left(\frac{j}{N_b - 1}\right)\right) = \left(\frac{j}{N_b - 1}, \frac{1}{1 - \lambda} \cos\left(\frac{2\pi j}{N_b - 1}\right)\right)$$

is also the fixed point of the map $T_j$ introduced in Proposition 2.2 page 9.

**Property 2.9.**

For $0 \le j \le \frac{(N_b - 1)}{2}$ (resp., for $\frac{(N_b - 1)}{2} \le j \le N_b - 1$), we have that

$$W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \le 0 \quad (\text{resp., } W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \ge 0).$$

*Proof.* For any integer $j$ in $\{0, \dots, N_b - 1\}$,

$$W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) = \frac{1}{1-\lambda}\left(\cos\left(\frac{2\pi(j+1)}{N_b-1}\right) - \cos\left(\frac{2\pi j}{N_b-1}\right)\right).$$

i. For $0 \le j \le \frac{N_b - 1}{2}$:

$$0 \le \frac{2\pi j}{N_b - 1} \le \pi, \quad 0 \le \frac{2\pi(j+1)}{N_b - 1} \le \pi \left(1 + \frac{2}{N_b - 1}\right).$$
<figure><img src="image_8.png" /><figcaption>Fig. 5. The polygonal neighborhood $\mathcal{D}(\Gamma_2)$, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. The polygonal neighborhood $\mathcal{D}(\Gamma_3)$, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$.</figcaption></figure>

Given $m \in \mathbb{N}$, we call $\mathcal{D}(\Gamma_m)$ the $m^{th}$ polyhedral neighborhood (of the Weierstrass Curve $\Gamma_W$); see Fig. 5, on page 18, along with Fig. 6, on page 18.

One now requires, at a given step $m \in \mathbb{N}^*$, the exact coordinates of the vertices of the prefractal graph $\Gamma_m$, i.e., of the following set of points:

$$ (x_{j,m}, W(x_{j,m})) \quad , \quad 0 \le j \le \#V_m. $$

Thus far, they could not be found in the existing literature on the subject.

For this purpose, it is interesting to use the scaling properties of the Weierstrass function.

**Property 2.8 (Scaling properties of the Weierstrass function, and consequences).** Since, for any real number $x$, $W(x) = \sum_{n=0}^{\infty} \lambda^n \cos(2\pi N_b^n x)$, one also has
The limit case

$$ \frac{2\pi(j+1)}{N_b-1} = \pi \left(1 + \frac{2}{N_b-1}\right) $$

only occurs when the integer $N_b$ is odd, for the value $j = \frac{N_b-1}{2}$, and corresponds to the bottom vertex of the initial polygon $P_0$. In this case, one has

$$ W\left(\frac{N_b-1}{2}\right) = -\frac{1}{1-\lambda} $$

This case can thus be left aside.

One may therefore only consider the cases when $0 \le \frac{2\pi j}{N_b-1} \le \frac{2\pi(j+1)}{N_b-1} \le \pi$.

The cosine function being nonincreasing on $[0, \pi]$, one obtains the expected result:

$$ W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \le 0 $$

ii. For $\frac{(N_b - 1)}{2} \le j \le N_b - 1$:

$$ \pi \le \frac{2\pi j}{N_b - 1} \le 2\pi \quad , \quad \pi \left(1 + \frac{2}{N_b - 1}\right) \le \frac{2\pi(j+1)}{N_b - 1} \le \frac{2\pi N_b}{N_b - 1} $$

As previously, the limit case

$$ \frac{2\pi(j+1)}{N_b-1} = \pi \left(1 + \frac{2}{N_b-1}\right) $$

can be left aside. The increasing property of the cosine function on $[\pi, 2\pi]$ then yields the expected result:

$$ W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \ge 0 $$

□

**Notation 7 (Signum Function).**

The *signum function* of a real number $x$ is defined by

$$ \operatorname{sgn}(x) = \begin{cases} -1, & \text{if } x < 0, \\ 0, & \text{if } x = 0, \\ +1, & \text{if } x > 0. \end{cases} $$
$$ 
\begin{aligned}
W(N_b x) &= \sum_{n=0}^{\infty} \lambda^n \cos (2\pi N_b^{n+1} x) = \frac{1}{\lambda} \sum_{n=1}^{\infty} \lambda^n \cos (2\pi N_b^n x) \\
&= \frac{1}{\lambda} (W(x) - \cos (2\pi x)) ,
\end{aligned}
$$

which yields, for any strictly positive integer $m$ and any $j$ in $\{0, \dots, \#V_m - 1\}$,

$$ W(x_{j,m}) = \lambda W(x_{j,m-1}) + \cos (2\pi x_{j,m}) . $$

By induction, one then obtains that

$$ W(x_{j,m}) = \lambda^m W\left(\frac{j}{N_b - 1}\right) + \sum_{k=0}^{m-1} \lambda^k \cos (2\pi N_b^k x_{j,m}) . $$

**Definition 2.7 (Left-side and right-side vertices).** Given nonnegative integers $m, k$ such that $0 \le k \le N_b^m - 1$, and a polygon $P_{m,k}$, we define:

i. The set of its *left-side vertices* as the set of the first $\lfloor \frac{N_b - 1}{2} \rfloor$ vertices, where $[y]$ denotes the integer part of the real number $y$.

ii. The set of its *right-side vertices* as the set of the last $\lfloor \frac{N_b - 1}{2} \rfloor$ vertices.

When the integer $N_b$ is odd, we define the bottom vertex as the $(\frac{N_b - 1}{2})^{th}$ one; see Fig. 8, on page 20.

**Property 2.9 (Exact values for the vertical coordinates of the vertices).** Since, for any nonnegative integer $n$,

$$ N_b^n = (1 + N_b - 1)^n = \sum_{k=0}^{n} \binom{n}{k} (N_b - 1)^k \equiv 1 \quad \text{mod } N_b - 1, $$

one obtains, for any integer $j$ in $\{0, \dots, N_b - 1\}$:

$$ 
\begin{aligned}
W\left(\frac{j}{N_b - 1}\right) &= \sum_{n=0}^{\infty} \lambda^n \cos \left(2\pi N_b^n \frac{j}{N_b - 1}\right) = \sum_{n=0}^{\infty} \lambda^n \cos \left(\frac{2\pi j}{N_b - 1}\right) \\
&= \frac{1}{1 - \lambda} \cos \left(\frac{2\pi j}{N_b - 1}\right) .
\end{aligned}
$$

We observe that the point

$$ \left( \frac{j}{N_b - 1}, W\left( \frac{j}{N_b - 1} \right) \right) = \left( \frac{j}{N_b - 1}, \frac{1}{1 - \lambda} \cos \left( \frac{2\pi j}{N_b - 1} \right) \right) $$
**Property 2.10.** Given any strictly positive integer $m$, we have the following properties:

i. For any $j$ in $\{0, \dots, \#V_m\}$, the point

$$ \left( \frac{j}{(N_b - 1) N_b^m}, W\left( \frac{j}{(N_b - 1) N_b^m} \right) \right) $$

is the image of the point

$$ \left( \frac{j}{(N_b - 1) N_b^{m-1}} - i, W\left( \frac{j}{(N_b - 1) N_b^{m-1}} - i \right) \right) = \left( \frac{j - i (N_b - 1) N_b^{m-1}}{(N_b - 1) N_b^{m-1}}, W\left( \frac{j - i (N_b - 1) N_b^{m-1}}{(N_b - 1) N_b^{m-1}} \right) \right) $$

under the map $T_i$, $0 \le i \le N_b - 1$.

Consequently, for $0 \le j \le N_b - 1$, the $j^{th}$ vertex of the polygon $P_{m,k}$, $0 \le k \le N_b^m - 1$, i.e., the point

$$ \left( \frac{(N_b - 1)k + j}{(N_b - 1) N_b^m}, W\left( \frac{(N_b - 1)k + j}{(N_b - 1) N_b^m} \right) \right) $$

is the image of the point

$$ \left( \frac{(N_b - 1) \left( k - i (N_b - 1) N_b^{m-1} \right) + j}{(N_b - 1) N_b^{m-1}}, W\left( \frac{(N_b - 1) \left( k - i (N_b - 1) N_b^{m-1} \right) + j}{(N_b - 1) N_b^{m-1}} \right) \right); $$

it is also the $j^{th}$ vertex of the polygon $P_{m-1,k-i(N_b-1)N_b^{m-1}}$. Therefore, there is an exact correspondance between vertices of the polygons at consecutive steps $m-1, m$.

ii. Given $j$ in $\{0, \dots, N_b - 2\}$ and $k$ in $\{0, \dots, N_b^m - 1\}$, we have that

$$ \operatorname{sgn}\left( W\left( \frac{k(N_b - 1) + j + 1}{(N_b - 1) N_b^m} \right) - W\left( \frac{k(N_b - 1) + j}{(N_b - 1) N_b^m} \right) \right) = \operatorname{sgn}\left( W\left( \frac{j + 1}{N_b - 1} \right) - W\left( \frac{j}{N_b - 1} \right) \right). $$

*Proof.*

i. One simply applies Proposition 2.3, on page 10, in conjunction with Property 2.8, on page 18.
<figure><img src="image_10.png" /><figcaption>Fig. 7. Symmetric points with respect to the vertical line $x = \frac{1}{2}$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. The left-side and right-side vertices.</figcaption></figure>

is also the fixed point of the map $T_j$ introduced in Proposition 2.2, on page 10.

**Property 2.10.** For $0 \le j \le \frac{(N_b - 1)}{2}$ (resp., for $\frac{(N_b - 1)}{2} \le j \le N_b - 1$), we have that

$$W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \le 0 \quad (\text{resp., } W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \ge 0).$$

The (rather technical) proof of Property 2.10 is provided in Appendix A.1, on page 62.
For $i$ in $\{0, \dots, N_b - 1\}$, we have that

$$
\begin{align*}
& T_i \left( \frac{j - i (N_b - 1) N_b^{m-1}}{(N_b - 1) N_b^{m-1}}, \mathcal{W} \left( \frac{j - i (N_b - 1) N_b^{m-1}}{(N_b - 1) N_b^{m-1}} \right) \right) \\
& \quad \left( \frac{j - i (N_b - 1) N_b^{m-1}}{(N_b - 1) N_b^m} + \frac{i}{N_b}, \lambda \mathcal{W} \left( \frac{j - i (N_b - 1) N_b^{m-1}}{(N_b - 1) N_b^{m-1}} \right) + \cos \left( 2 \pi \left( \frac{j - i (N_b - 1) N_b^{m-1}}{(N_b - 1) N_b^m} + \frac{i}{N_b} \right) \right) \right) \\
&= \left( \frac{j}{(N_b - 1) N_b^m}, \mathcal{W} \left( \frac{j}{(N_b - 1) N_b^{m-1}} - i \right) + \cos \left( 2 \pi \frac{j}{(N_b - 1) N_b^m} \right) \right) \\
&= \left( \frac{j}{(N_b - 1) N_b^m}, \mathcal{W} \left( \frac{j}{(N_b - 1) N_b^{m-1}} - i \right) + \cos \left( 2 \pi \frac{j - i}{(N_b - 1) N_b^m} + \frac{i}{N_b} \right) \right) \\
&= \left( \frac{j}{(N_b - 1) N_b^m}, \lambda \mathcal{W} \left( \frac{j}{(N_b - 1) N_b^{m-1}} \right) + \cos \left( 2 \pi \frac{j - i}{(N_b - 1) N_b^m} \right) \right) \\
&= \left( \frac{j}{(N_b - 1) N_b^m}, \mathcal{W} \left( \frac{j}{(N_b - 1) N_b^m} \right) \right).
\end{align*}
$$

ii. We prove the result by induction on $m$. Accordingly, let us consider $j$ in $\{0, \dots, N_b - 2\}$.

The result at the initial step $m=1$ is satisfied, in so far as, for any integer $k$ in $\{0, \dots, N_b - 1\}$:

$$
\begin{align*}
\mathcal{W}\left(\frac{k(N_b - 1) + j + 1}{(N_b - 1) N_b}\right) - \mathcal{W}\left(\frac{k(N_b - 1) + j}{(N_b - 1) N_b}\right) &= \lambda \left(\mathcal{W}\left(\frac{k(N_b - 1) + j + 1}{N_b - 1}\right) - \mathcal{W}\left(\frac{k(N_b - 1) + j}{N_b - 1}\right)\right) \\
&\quad + \cos\left(\frac{2\pi(k(N_b - 1) + j + 1)}{N_b - 1}\right) - \cos\left(\frac{2\pi(k(N_b - 1) + j)}{N_b - 1}\right) \\
&= \lambda \left(\mathcal{W}\left(k + \frac{j+1}{N_b-1}\right) - \mathcal{W}\left(k + \frac{j}{N_b-1}\right)\right) \\
&\quad + \cos\left(\frac{2\pi(j+1)}{N_b-1}\right) - \cos\left(\frac{2\pi j}{N_b-1}\right) \\
&= \lambda \left(\mathcal{W}\left(\frac{j+1}{N_b-1}\right) - \mathcal{W}\left(\frac{j}{N_b-1}\right)\right) \\
&\quad + \mathcal{W}\left(\frac{j+1}{N_b-1}\right) - \mathcal{W}\left(\frac{j}{N_b-1}\right) \\
&= (1 + \lambda) \left(\mathcal{W}\left(\frac{j+1}{N_b-1}\right) - \mathcal{W}\left(\frac{j}{N_b-1}\right)\right).
\end{align*}
$$

Let us now assume that, for any integer $k$ in $\{0, \dots, N_b^{m-1} - 1\}$,

$$
\operatorname{sgn}\left(\mathcal{W}\left(\frac{k(N_b-1)+j+1}{(N_b-1)N_b^m}\right)-\mathcal{W}\left(\frac{k(N_b-1)j}{(N_b-1)N_b^m}\right)\right)=\operatorname{sgn}\left(\mathcal{W}\left(\frac{j+1}{N_b-1}\right)-\mathcal{W}\left(\frac{j}{N_b-1}\right)\right).
$$
**Notation 4 (Signum function).** The signum function of a real number $x$ is defined by

$$ \operatorname{sgn}(x) = \begin{cases} -1, & \text{if } x < 0, \\ 0, & \text{if } x = 0, \\ +1, & \text{if } x > 0. \end{cases} $$

**Property 2.11.** Given any strictly positive integer $m$, we have the following properties:

i. For any $j$ in $\{0, \dots, \#V_m - 1\}$, the point

$$ (x_{j,m}, \mathcal{W}(x_{j,m})) $$

is the image of the point

$$ (x_{j,m-1} - i, \mathcal{W}(x_{j,m-1} - i)) = (x_{j-i(N_b-1)N_b^{m-1}, m-1}, \mathcal{W}(x_{j-i(N_b-1)N_b^{m-1}, m-1})) $$

under the map $T_i$, $0 \le i \le N_b - 1$.

Consequently, for $0 \le j \le N_b - 1$, the $j^{th}$ vertex of the polygon $\mathcal{P}_{m,k}$, $0 \le k \le N_b^m - 1$, i.e., the point

$$ (x_{(N_b-1)k+j,m}, \mathcal{W}(x_{(N_b-1)k+j,m})) $$

is the image of the point

$$ \left( x_{(N_b-1)(k-i(N_b-1)N_b^{m-1})+j,m-1}, \mathcal{W}\left(x_{(N_b-1)(k-i(N_b-1)N_b^{m-1})+j,m-1}\right) \right), $$

which is also the $j^{th}$ vertex of the polygon $\mathcal{P}_{m-1,k-i(N_b-1)N_b^{m-1}}$. Therefore, there is an exact correspondence between vertices of the polygons at consecutive steps $m-1, m$.

ii. Given $j$ in $\{0, \dots, N_b - 2\}$ and $k$ in $\{0, \dots, N_b^m - 1\}$, we have that

$$ \begin{aligned} & \operatorname{sgn}\left(\mathcal{W}\left(x_{k(N_b-1)+j+1,m}\right) - \mathcal{W}\left(x_{k(N_b-1)+j,m}\right)\right) \\ & = \operatorname{sgn}\left(\mathcal{W}\left(\frac{j+1}{N_b-1}\right) - \mathcal{W}\left(\frac{j}{N_b-1}\right)\right). \end{aligned} $$

The (very technical) proof of Property 2.11 is given in Appendix A.2, on page 63.

**Corollary 2.12** (Lower bound for the elementary heights (coming from Property 2.11, on page 21)). For any strictly positive integer $m$, and any $j$ in $\{0, \dots, (N_b - 1) N_b^m\}$, we have that
Henceforth, we want to prove that, for any integer $k$ in $\{0, \dots, N_b^{m-1} - 1\}$,

$$ \operatorname{sgn}\left(\mathcal{W}\left(\frac{k(N_b-1)+j+1}{(N_b-1)N_b^m}\right)-\mathcal{W}\left(\frac{k(N_b-1)j}{(N_b-1)N_b^m}\right)\right)=\operatorname{sgn}\left(\mathcal{W}\left(\frac{j+1}{N_b-1}\right)-\mathcal{W}\left(\frac{j}{N_b-1}\right)\right). $$

The induction hypothesis will be used in so far as any $k$ in $\{0, \dots, N_b^{m-1} - 1\}$ can also be expressed in the following form:

$$ k = \tilde{k} + i N_b^{m-1}, \quad 0 \le \tilde{k} \le N_b^{m-1} - 1, \quad 0 \le i \le N_b - 1. $$

This will be useful because of the one-periodicity of the $W$-function, since, for any real number $x$ and any integer $i$, we have that

$$ \mathcal{W}(x + i) = \mathcal{W}(x). $$

Due to the symmetry with respect to the vertical line $x = \frac{1}{2}$ (see Property 2.1, on page 9), given a natural integer $m$, one can, in addition, restrict oneself to the cases when

$$ 0 \le (N_b - 1)k + j < (N_b - 1)k + j + 1 \le \left[ \frac{(N_b - 1)N_b^m + 1}{2} \right] = \frac{(N_b - 1)N_b^m}{2}, $$

which yields

$$ 0 \le \frac{(2(N_b - 1)k + 2j - 1)\pi}{2(N_b - 1)N_b^m} < \frac{(2(N_b - 1)k + 2j + 1)\pi}{(N_b - 1)N_b^m} \le \pi. $$

Thus, we only have to consider the cases when

$$ \sin\left(\frac{(2(N_b - 1)k + 2j - 1)\pi}{(N_b - 1)N_b^m}\right) \ge 0 \quad \text{and} \quad \sin\left(\frac{(2(N_b - 1)k + 2j + 1)\pi}{(N_b - 1)N_b^m}\right) \ge 0. $$

The remaining ones, namely, the cases when

$$ \sin\left(\frac{(2(N_b - 1)k + 2j - 1)\pi}{(N_b - 1)N_b^m}\right) \le 0 \quad \text{and} \quad \sin\left(\frac{(2(N_b - 1)k + 2j + 1)\pi}{(N_b - 1)N_b^m}\right) \le 0, $$

are then obtained by symmetry.

Hence,

$$ \mathcal{W}\left(\frac{k(N_b - 1) + j + 1}{(N_b - 1)N_b^m}\right) - \mathcal{W}\left(\frac{j}{(N_b - 1)N_b^m}\right) \quad \| $$
$$|W(x_{j+1,m}) - W(x_{j,m})| \ge \lambda |W(x_{j+1,m-1}) - W(x_{j,m-1})|,$$

which yields, by induction,

$$|W(x_{j+1,m}) - W(x_{j,m})| \ge \underbrace{\lambda^m}_{N_b^m(D_W-2)} \left| W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \right|.$$

This improves our previous result in [4].

**Corollary 2.13** (Upper bound for the elementary heights (coming from Property 2.11, on page 21)). For any strictly positive integer $m$, and any $j$ in $\{0, \dots, (N_b - 1) N_b^m\}$, we have that

$$|W(x_{j+1,m}) - W(x_{j,m})| \\ \le \lambda^m \left( \left| W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \right| + \frac{2\pi}{(N_b-1)(\lambda N_b - 1)} \right) \\ \le \underbrace{\lambda^m}_{N_b^m(D_W-2)} \left( \left| W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \right| + \frac{2\pi}{(N_b-1)(\lambda N_b - 1)} \right),$$

which also improves our previous result in [4].

**Proof.** For any strictly positive integer $m$ and any $j$ in $\{0, \dots, (N_b - 1) N_b^m\}$, we have the following estimates:

$$|W(x_{j+1,m}) - W(x_{j,m})| \le \lambda |W(x_{j+1,m-1}) - W(x_{j,m-1})| \\ + |\cos(2\pi x_{j+1,m-1}) - \cos(2\pi x_{j,m-1})| \\ \le \lambda |W(x_{j+1,m-1}) - W(x_{j,m-1})| + \frac{2\pi}{(N_b-1)N_b^{m-1}},$$

which yields, by induction,

$$|W(x_{j+1,m}) - W(x_{j,m})| \le \lambda^m \left| W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \right| \\ + \sum_{k=0}^{m-1} \lambda^k \frac{2\pi N_b^k}{(N_b-1)N_b^m} \\ \le \lambda^m \left| W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \right| \\ + \frac{2\pi \lambda^m N_b^m}{(N_b-1)N_b^m (\lambda N_b - 1)}$$
$$ = \lambda \left( W \left( \frac{k(N_b - 1) + j + 1}{(N_b - 1) N_b^{m-1}} \right) - W \left( \frac{k(N_b - 1) + j}{(N_b - 1) N_b^{m-1}} \right) \right) \\ + \cos \left( \frac{2\pi (k(N_b - 1) + j + 1)}{(N_b - 1) N_b^{m-1}} \right) - \cos \left( \frac{2\pi (k(N_b - 1) + j)}{(N_b - 1) N_b^{m-1}} \right) $$

$$ = \lambda \left( W \left( \frac{k(N_b - 1) + j + 1}{(N_b - 1) N_b^{m-1}} \right) - W \left( \frac{k(N_b - 1) + j}{(N_b - 1) N_b^{m-1}} \right) \right) \\ - 2 \sin \left( \frac{\pi}{(N_b - 1) N_b^{m-1}} \right) \sin \left( \frac{(2(N_b - 1)k + 2j + 1)\pi}{(N_b - 1) N_b^{m-1}} \right) $$

$$ = \lambda \left( W \left( \frac{\tilde{k}(N_b - 1) + i(N_b - 1) N_b^{m-1} + j + 1}{(N_b - 1) N_b^{m-1}} \right) - W \left( \frac{\tilde{k}(N_b - 1) + i(N_b - 1) N_b^{m-1} + j}{(N_b - 1) N_b^{m-1}} \right) \right) \\ - 2 \sin \left( \frac{\pi}{(N_b - 1) N_b^{m-1}} \right) \sin \left( \frac{(2(N_b - 1)k + 2j + 1)\pi}{(N_b - 1) N_b^{m-1}} \right) $$

$$ = \lambda \left( W \left( i + \frac{\tilde{k}(N_b - 1) + j + 1}{(N_b - 1) N_b^{m-1}} \right) - W \left( i + \frac{\tilde{k}(N_b - 1) + j}{(N_b - 1) N_b^{m-1}} \right) \right) \\ - 2 \sin \left( \frac{\pi}{(N_b - 1) N_b^{m-1}} \right) \sin \left( \frac{(2(N_b - 1)k + 2j + 1)\pi}{(N_b - 1) N_b^{m-1}} \right) $$

$$ = \lambda \left( W \left( \frac{\tilde{k}(N_b - 1) + j + 1}{(N_b - 1) N_b^{m-1}} \right) - W \left( \frac{\tilde{k}(N_b - 1) + j}{(N_b - 1) N_b^{m-1}} \right) \right) \\ - 2 \sin \left( \frac{\pi}{(N_b - 1) N_b^{m-1}} \right) \sin \left( \frac{(2(N_b - 1)k + 2j + 1)\pi}{(N_b - 1) N_b^{m-1}} \right) . $$

In the case when

$$ 0 \le (N_b - 1)k + j + 1 \le \left[ \frac{(N_b - 1)N_b^m + 1}{2} \right] = \frac{(N_b - 1)N_b^m}{2}, $$

one thus has

$$ -2 \sin \left( \frac{\pi}{(N_b - 1) N_b^{m-1}} \right) \sin \left( \frac{(2(N_b - 1)k + 2j - 1)\pi}{(N_b - 1) N_b^{m-1}} \right) \le 0. $$

The configuration of the initial polygon ensures, for $0 \le j \le \frac{N_b - 1}{2}$, that

$$ W \left( \frac{j+1}{N_b-1} \right) - W \left( \frac{j}{N_b-1} \right) \le 0 $$

and therefore, thanks to the induction hypothesis,

$$ W \left( \frac{\tilde{k}(N_b - 1) + j + 1}{(N_b - 1) N_b^{m-1}} \right) - W \left( \frac{\tilde{k}(N_b - 1) + j}{(N_b - 1) N_b^{m-1}} \right) \le 0. $$
$$ = \lambda^m \left( \left| W \left( \frac{j+1}{N_b-1} \right) - W \left( \frac{j}{N_b-1} \right) \right| + \frac{2\pi}{(N_b-1)(\lambda N_b - 1)} \right), $$

as desired. □

**Remark 2.3.** Corollaries 2.12 (page 21) and 2.13 (page 22) are important because they enable one to obtain exact and more accurate values of the bounding constants $C_{inf}$ and $C_{sup}$ involved in the following inequality:

$$ C_{inf} L_m^{2-D_W} \le \underbrace{|W(x_{j+1,m}) - W(x_{j,m})|}_{h_{j,m}} \le C_{sup} L_m^{2-D_W}, \quad m \in \mathbb{N}, 0 \le j \le (N_b-1) N_b^m, \quad (6) $$

where

$$ C_{inf} = (N_b - 1)^{2-D_W} \min_{0 \le j \le N_b - 1, W(\frac{j+1}{N_b-1}) \ne W(\frac{j}{N_b-1})} \left| W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \right| $$

and

$$ C_{sup} = (N_b - 1)^{2-D_W} \left( \max_{0 \le j \le N_b - 1} \left| W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \right| + \frac{2\pi}{(N_b-1)(\lambda N_b - 1)} \right). $$

One should note, in addition, that these constants depend on the initial polygon $P_0$.

**Theorem 2.14** (Sharp local discrete reverse Hölder properties of the Weierstrass function (coming from Corollary 2.12, on page 21)). For any nonnegative integer $m$, let us consider a pair of real numbers $(x, x')$ such that

$$ x = \frac{(N_b - 1)k + j}{(N_b - 1)N_b^m} = ((N_b - 1)k + j) L_m, $$

$$ x' = \frac{(N_b - 1)k + j + \ell}{(N_b - 1)N_b^m} = ((N_b - 1)k + j + \ell) L_m, $$

where $0 \le k \le N_b^m - 1$, and

i. if the integer $N_b$ is odd,

$$ 0 \le j < \frac{N_b - 1}{2} \quad \text{and} \quad 0 < j + \ell \le \frac{N_b - 1}{2} $$

or
By induction, one thus obtains, for any natural integer $m$, any $k$ in $\{0, \dots, N_b^m - 1\}$, and any $j$ in $\{0, \dots, \frac{N_b - 3}{2}\}$, that

$$W\left(\frac{(N_b - 1)k + j + 1}{(N_b - 1)N_b^m}\right) - W\left(\frac{(N_b - 1)k + j}{(N_b - 1)N_b^m}\right) \le 0,$$

as required. □

**Corollary 2.11 (Lower Bound for the Elementary Heights (Coming from Property 2.10, on page 20)).**

For any strictly positive integer $m$, and any $j$ in $\{0, \dots, (N_b - 1)N_b^m\}$, we have that

$$\left|W\left(\frac{j+1}{(N_b-1)N_b^m}\right) - W\left(\frac{j}{(N_b-1)N_b^m}\right)\right| \ge \lambda \left|W\left(\frac{j+1}{(N_b-1)N_b^{m-1}}\right) - W\left(\frac{j}{(N_b-1)N_b^{m-1}}\right)\right|,$$

which yields, by induction,

$$\left|W\left(\frac{j+1}{(N_b-1)N_b^m}\right) - W\left(\frac{j}{(N_b-1)N_b^m}\right)\right| \ge \lambda^{m} \sum_{N_b^m(D_W-2)} \left|W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right)\right|.$$

This improves our previous result in [Dav18].

**Corollary 2.12 (Upper Bound for the Elementary Heights (Coming from Property 2.10, on page 20)).**

For any strictly positive integer $m$, and any $j$ in $\{0, \dots, (N_b - 1)N_b^m\}$, we have that

$$\left|W\left(\frac{j+1}{(N_b-1)N_b^m}\right) - W\left(\frac{j}{(N_b-1)N_b^m}\right)\right| \le \lambda^m \left(\left|W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right)\right| + \frac{2\pi}{(N_b-1)(\lambda N_b - 1)}\right) \\ \le \lambda^m \sum_{N_b^m(D_W-2)} \left(\left|W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right)\right| + \frac{2\pi}{(N_b-1)(\lambda N_b - 1)}\right),$$

which also improves our previous result in [Dav18].
$$ \frac{N_b - 1}{2} \le j < N_b - 1 \quad \text{and} \quad \frac{N_b - 1}{2} < j + \ell \le N_b - 1; $$

ii. if the integer $N_b$ is even,

$$ 0 \le j < \frac{N_b}{2} \quad \text{and} \quad 0 < j + \ell \le \frac{N_b}{2} $$

or

$$ \frac{N_b}{2} + 1 \le j < N_b - 1 \quad \text{and} \quad \frac{N_b}{2} + 1 < j + \ell \le N_b - 1. $$

This means that the points $(x, W(x))$ and $(x', W(x'))$ are vertices of the polygon $P_{m,k}$ (see part iii. of Property 2.7, on page 15 above), both located on the left-side of the polygon, or both located on the right-side; see Fig. 8, on page 20.

Then, one has the following (discrete, local) reverse-Hölder inequality, with sharp Hölder exponent $-\frac{\ln \lambda}{\ln N_b} = 2 - D_W$,

$$ C_{inf} |x' - x|^{2-D_W} \le |W(x') - W(x)|. $$

**Proof.** In the light of Property 2.10, on page 20, one can restrict oneself to the case when

$$ 0 \le j < \frac{N_b - 1}{2} \quad \text{and} \quad 0 < j + \ell \le \frac{N_b - 1}{2}. $$

The expected result in the remaining case can easily be proved in a similar way. Since

$$ W(x_{(N_b-1)k+j+\ell,m}) \le \dots \le W(x_{(N_b-1)k+j+1,m}) \le W(x_{(N_b-1)k+j,m}), $$

then, by applying the results of Remark 2.3, on page 23, we have, for $0 \le i \le \ell - 1$, the following $\ell$ inequalities:

$$ C_{inf} L_m^{2-D_W} \le -W(x_{(N_b-1)k+j+i+1,m}) + W(x_{(N_b-1)k+j+i,m}). $$

Thus, upon summation, we obtain that

$$ \ell C_{inf} L_m^{2-D_W} \le -W(x_{(N_b-1)k+j+\ell,m}) + W(x_{(N_b-1)k+j,m}). $$

Since $\ell \ge \ell^{2-D_W}$ and $|x' - x| = \ell L_m$, one deduces the desired result. $\square$

**Remark 2.4.** Thus far, no such reverse Hölder estimates had been obtained for the Weierstrass function. The fact that they are discrete ones is natural, since the Weierstrass Curve is approximated by a sequence of polygonal prefractal finite graphs. Recall that the countable set of vertices of all of these graphs is dense in the whole Weierstrass Curve.
Proof. For any strictly positive integer $m$ and any $j$ in $\{0, \dots, (N_b - 1) N_b^m\}$, we have the following estimates:

$$ \left| W\left(\frac{j+1}{(N_b-1)N_b^m}\right) - W\left(\frac{j}{(N_b-1)N_b^m}\right) \right| \le \lambda \left| W\left(\frac{j+1}{(N_b-1)N_b^{m-1}}\right) - W\left(\frac{j}{(N_b-1)N_b^{m-1}}\right) \right| + \left| \cos\left(\frac{2\pi(j+1)}{(N_b-1)N_b^{m-1}}\right) - \cos\left(\frac{2\pi j}{(N_b-1)N_b^{m-1}}\right) \right| \\ \le \lambda \left| W\left(\frac{j+1}{(N_b-1)N_b^{m-1}}\right) - W\left(\frac{j}{(N_b-1)N_b^{m-1}}\right) \right| + \frac{2\pi}{(N_b-1)N_b^{m-1}}, $$

which yields, by induction,

$$ \begin{aligned} \left| W\left(\frac{j+1}{(N_b-1)N_b^m}\right) - W\left(\frac{j}{(N_b-1)N_b^m}\right) \right| &\le \lambda^m \left| W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \right| + \sum_{k=0}^{m-1} \lambda^k \frac{2\pi N_b^k}{(N_b-1)N_b^m} \\ &\le \lambda^m \left| W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \right| + \frac{2\pi \lambda^m N_b^m}{(N_b-1)N_b^m (\lambda N_b - 1)} \\ &= \lambda^m \left( \left| W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \right| + \frac{2\pi}{(N_b-1)(\lambda N_b - 1)} \right), \end{aligned} $$

as desired. $\square$

Remark 2.4. Corollaries 2.11 (page 24) and 2.12 (page 24) are important, because they enable one to obtain exact and more accurate values of the bounding constants $C_{inf}$ and $C_{sup}$ involved in the following inequality:

$$ C_{inf} L_m^{2-D_W} \le \underbrace{|W((j+1)L_m) - W(jL_m)|}_{h_{j,j+1,m}} \le C_{sup} L_m^{2-D_W}, \quad m \in \mathbb{N}, 0 \le j \le (N_b-1)N_b^m, \quad (\star) \tag{R 7} $$

where

$$ C_{inf} = (N_b - 1)^{2-D_W} \min_{0 \le j \le N_b - 1} \left| W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \right| $$

and

$$ C_{sup} = (N_b - 1)^{2-D_W} \left( \max_{0 \le j \le N_b - 1} \left| W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \right| + \frac{2\pi}{(N_b-1)(\lambda N_b - 1)} \right). $$

One should note, in addition, that these constants depend on the initial polygon $\mathcal{P}_0$.
**Corollary 2.15 (Optimal Hölder exponent for the Weierstrass function).** The local reverse Hölder property of Theorem 2.14, on page 23 – in conjunction with the Hölder condition satisfied by the Weierstrass function (see also [56], Chapter II, Theorem 4.9, page 47) – shows that the Codimension $2 - D_W = -\frac{\ln \lambda}{\ln N_b} \in ]0, 1[$ is the best (i.e., optimal) Hölder exponent for the Weierstrass function (as was originally shown, by a completely different method, by G. H. Hardy in [17]).

Note that, as a consequence, since the Hölder exponent is strictly smaller than one, the Weierstrass function $W$ is nowhere differentiable.

**Remark 2.5.** Indeed, if $W$ were differentiable at some point $x_0 \in [0, 1]$, then it would have to be locally Lipschitz at $x_0$, and hence, its Hölder exponent at $x_0$ would be equal to 1, which is impossible.

**Corollary 2.16 (Coming from Property 2.11, on page 21).** Thanks to Remark 2.3, on page 23, one may now write, for any strictly positive integer $m$ and any integer $j$ in $\{1, \dots, (N_b - 1) N_b^m - 1\}$:

i. For the elementary heights:

$$C_{inf} L_m^{2-D_W} \le h_{j-1,m} \le C_{sup} L_m^{2-D_W}, \quad (7)$$

or, equivalently,

$$h_{j-1,m} = C_{h,j-1,m} L_m^{2-D_W}, \quad (8)$$

where $C_{h,j-1,m}$ depends on $j$ and $m$ and is such that

$$C_{inf} \le C_{h,j-1,m} \le C_{sup}.$$

ii. For the elementary lengths:

$$\sqrt{C_{inf}^2 + 1} L_m^{2-D_W} \le \ell_{j-1,m} \le \sqrt{C_{sup}^2 + 1} L_m^{2-D_W}, \quad (9)$$

or, equivalently,

$$\ell_{j-1,m} = C_{\ell,j-1,m} L_m^{2-D_W}, \quad (10)$$

where $C_{\ell,j-1,m}$ depends on $j$ and $m$ and is such that

$$\sqrt{1 + C_{sup}^{-2}} C_{inf} \le C_{\ell,j-1,m} \le \sqrt{1 + C_{inf}^{-2}} C_{sup}.$$
**Theorem 2.13 (Sharp Local Discrete Reverse Hölder Properties of the Weierstrass Function (Coming from Corollary 2.11, on page 24)).**

For any natural integer $m$, let us consider a pair of real numbers $(x, x')$ such that

$$x = \frac{(N_b - 1)k + j}{(N_b - 1)N_b^m} = ((N_b - 1)k + j) L_m, \quad x' = \frac{(N_b - 1)k + j + \ell}{(N_b - 1)N_b^m} = ((N_b - 1)k + j + \ell) L_m,$$

where $0 \le k \le N_b - 1^m - 1$, and

i. if the integer $N_b$ is odd,

$$0 \le j < \frac{N_b - 1}{2} \quad \text{and} \quad 0 < j + \ell \le \frac{N_b - 1}{2}$$

or

$$\frac{N_b - 1}{2} \le j < N_b - 1 \quad \text{and} \quad \frac{N_b - 1}{2} < j + \ell \le N_b - 1;$$

ii. if the integer $N_b$ is even,

$$0 \le j < \frac{N_b}{2} \quad \text{and} \quad 0 < j + \ell \le \frac{N_b}{2}$$

or

$$\frac{N_b}{2} + 1 \le j < N_b - 1 \quad \text{and} \quad \frac{N_b}{2} + 1 < j + \ell \le N_b - 1.$$

This means that the points $(x, W(x))$ and $(x', W(x'))$ are vertices of the polygon $P_{m,k}$ (see Property 2.4, on page 14 above), both located on the left-side of the polygon, or both located on the right-side; see Figure 6, on page 18.

Then, one has the following (discrete, local) reverse-Hölder inequality, with sharp Hölder exponent $-\frac{\ln \lambda}{\ln N_b} = 2 - D_W$,

$$C_{inf} |x' - x|^{2-D_W} \le |W(x') - W(x)|.$$

*Proof.* In the light of Property 2.9, on page 18, one can restrict oneself to the case when

$$0 \le j < \frac{N_b - 1}{2} \quad \text{and} \quad 0 < j + \ell \le \frac{N_b - 1}{2}.$$

The expected result in the remaining case can easily be proved in a similar way. Since

$$W(((N_b - 1)k + j + \ell) L_m) \le \dots \le W(((N_b - 1)k + j + 1) L_m) \le W(((N_b - 1)k + j) L_m)$$

then, by applying the results of Remark 2.4, on page 25, we have the following $\ell$ inequalities:

$$C_{inf} L_m^{2-D_W} \le -W(((N_b - 1)k + j + 1) L_m) + W((N_b - 1)k + j) L_m)$$

$$C_{inf} L_m^{2-D_W} \le -W(((N_b - 1)k + j + 2) L_m) + W(((N_b - 1)k + j + 1) L_m)$$
**Proof.** We simply use the fact that

$$ \ell_{j-1,m} = \sqrt{h_{j-1,m}^2 + L_m^2} \\ = h_{j-1,m} \sqrt{1 + C_{h,j-1,m}^{-2} L_m^{D_W}}, $$

where

$$ 0 < L_m^{D_W} < 1 \quad \text{and} \quad C_{sup}^{-2} \le C_{h,j-1,m}^{-2} \le C_{inf}^{-2}. $$

We thus obtain that

$$ \sqrt{1 + C_{sup}^{-2}} \le \sqrt{1 + C_{h,j-1,m}^{-2} L_m^{D_W}} \le \sqrt{1 + C_{inf}^{-2}}, $$

which ensures that

$$ \sqrt{1 + C_{sup}^{-2}} C_{inf} L_m^{2-D_W} \le \ell_{j-1,m} \le \sqrt{1 + C_{inf}^{-2}} C_{sup} L_m^{2-D_W}. \quad \square $$

**Corollary 2.17** (Nonincreasing sequence of geometric angles (coming from Property 2.11)). For the geometric angles $\theta_{j-1,m}$, $0 \le j \le (N_b - 1) N_b^m - 1$, $m \in \mathbb{N}$, we have the following result:

$$ \tan \theta_{j-1,m} = \frac{L_m}{h_{j-1,m}} (N_b - 1) > \tan \theta_{j-1,j,m+1}, $$

which yields

$$ \theta_{j-1,m} > \theta_{j-1,j,m+1} \quad \text{and} \quad \theta_{j-1,m+1} \lesssim L_m^{D_W-1}. $$

The (technical) proof of Corollary 2.17 is given in Appendix A.3, on page 67.

**Corollary 2.18** (Local extrema of the Weierstrass function (coming from Property 2.11, on page 21)).

i. The set of local maxima of the Weierstrass function on the interval $[0, 1]$ is given by

$$ \{(x_{k,m}, W(x_{k,m})) : 0 \le k \le N_b^m - 1, m \in \mathbb{N}\}, $$

and corresponds to the extreme vertices of the polygons at a given step $m$ (vertices connecting consecutive polygons).

ii. For odd values of $N_b$, the set of local minima of the Weierstrass function on the interval $[0, 1]$ is given by

$$ \{(x_{(N_b-1)k + \frac{N_b-1}{2}}, m), W(x_{(N_b-1)k + \frac{N_b-1}{2}}, m)) : 0 \le k \le N_b^m - 1, m \in \mathbb{N}\}, $$
$$C_{inf} L_m^{2-D_W} \le -W(((N_b - 1)k + j + \ell) L_m) + W(((N_b - 1)k + \ell - 1) L_m)$$

Thus, upon summation, we obtain that

$$\ell C_{inf} L_m^{2-D_W} \le -W(((N_b - 1)k + j + \ell) L_m) + W(((N_b - 1)k + j) L_m)$$

Since $\ell \ge \ell^{2-D_W}$ and $|x' - x| = \ell L_m$, one deduces the desired result.

□

**Remark 2.5.** Thus far, no such reverse Hölder estimates had been obtained for the Weierstrass function. The fact that they are discrete ones is natural, since the Weierstrass Curve is approximated by a sequence of polygonal prefractal finite graphs. Recall that the countable set of vertices of all of these graphs is dense in the whole Weierstrass Curve.

**Corollary 2.14 (Optimal Hölder Exponent for the Weierstrass Function).**

The local reverse Hölder property of Theorem 2.13, on page 26 – in conjunction with the Hölder condition satisfied by the Weierstrass function (see also [Zyg02], Chapter II, Theorem 4.9, page 47) – shows that the Codimension $2 - D_W = -\frac{\ln \lambda}{\ln N_b} \in ]0, 1[$ is the best (i.e., optimal) Hölder exponent for the Weierstrass function (as was originally shown, by a completely different method, by G. H. Hardy in [Har16]).

Note that, as a consequence, since the Hölder exponent is strictly smaller than one, the Weierstrass function $W$ is nowhere differentiable.

**Remark 2.6.** Indeed, if $W$ were differentiable at some point $x_0 \in [0, 1]$, then it would have to be locally Lipschitz at $x_0$, and hence, its Hölder exponent at $x_0$ would be equal to 1, which is impossible.

**Corollary 2.15 (Coming from Property 2.10, on page 20).**

Thanks to Remark 2.4, on page 25, one may now write, for any strictly positive integer $m$ and any integer $j$ in $\{0, \dots, (N_b - 1) N_b^m - 1\}$:

i. for the elementary heights:

$$h_{j-1,j,m} = L_m^{2-D_W} \mathcal{O}(1); \qquad (\mathcal{R} 8)$$

ii. for the elementary quotients:

$$\frac{h_{j-1,j,m}}{L_m} = L_m^{1-D_W} \mathcal{O}(1), \qquad (\mathcal{R} 9)$$

as follows from Remark 2.4, on page 25 above, and where

$$0 < C_{inf} \le \mathcal{O}(1) \le C_{sup}.$$
and corresponds to the bottom vertices of the polygons at a given step m.

**Property 2.19 (Existence of reentrant angles).**

i. The initial polygon $P_0$, admits **reentrant interior angles**, at a vertex $P_j$, with $0 < j \le N_b - 1$, in the sense that, with the right-hand rule (according to which angles are measured in a counterclockwise direction), we have that

$$((P_j P_{j+1}), (\widehat{P_j P_{j-1}})) > \pi,$$

for

$$0 < j \le \frac{N_b - 3}{4} \quad \text{or} \quad \frac{3N_b - 1}{4} \le j < N_b - 1$$

(see Fig. 9, on page 28), which does not occur for values of $N_b < 7$.

The number of reentrant angles is then equal to $2 \left[ \frac{N_b - 3}{4} \right]$.

ii. At a given step $m \in \mathbb{N}^*$, with the above convention, a polygon $P_{m,k}$ admits reentrant interior angles in the sole cases when $N_b \ge 7$, at vertices $M_{k+j}$, $1 \le k \le N_b^m$, $0 < j \le N_b - 1$, as well as when

$$0 < j \le \frac{N_b - 3}{4} \quad \text{or} \quad \frac{3N_b - 1}{4} \le j < N_b - 1.$$

The number of reentrant angles is then equal to $2 N_b^m \left[ \frac{N_b - 3}{4} \right]$.

The (technical) proof of Property 2.19 is provided in Appendix A.4, on page 68.

**Definition 2.8 (Self-shape similarity of the Weierstrass Curve).** We will say that the Weierstrass Curve – as the two-dimensional Hausdorff and uniform limit curve of a sequence of polygonal prefraçals, which satisfy Property 2.11, on page 21 and Property 2.19, on page 27 – has self-shape similarity, in the sense that the shape of the initial polygon $P_0$ governs the shape of all the polygons $P_{m,k}$, with $0 \le k \le N_b^m$, at any step $m$ of the prefraçal approximation process. This self-shape similarity property is apparent in Fig. 1, on page 12, in the case when $N_b = 3$, Fig. 2, on page 13, in the case when $N_b = 4$, and Fig. 3, on page 14, in the case when $N_b = 7$. As for the existence of reentrant angles, it can be observed in the first two graphs of Fig. 3, on page 14, in the case when $N_b = 7$.

**3. Iterated fractal drums and tubular neighborhoods**

In the case of classical fractals, and when the associated geometry allows it, the values of the Complex Dimensions are obtained by studying the oscillations of a small neighborhood of the boundary, i.e., of a tubular neighborhood of the fractal, where points
**Corollary 2.16 (Nonincreasing Sequence of Geometric Angles (Coming from Property 2.10)).**

For the geometric angles $\theta_{j-1,j,m}$, $0 \le j \le (N_b - 1) N_b^m - 1$, $m \in \mathbb{N}$, we have the following result:

$$ \tan \theta_{j-1,j,m} = \frac{L_m}{h_{j-1,j,m}} (N_b - 1) > \tan \theta_{j-1,j,m+1}, $$

which yields

$$ \theta_{j-1,j,m} > \theta_{j-1,j,m+1} \quad \text{and} \quad \theta_{j-1,j,m+1} \le L_m^{D_W-1}. $$

*Proof.*

i. One simply writes, successively:

$$ \begin{aligned} \tan \theta_{j-1,j,m} &= \frac{L_m}{\left| W\left(\frac{j}{(N_b - 1) N_b^m}\right) - W\left(\frac{j-1}{(N_b - 1) N_b^m}\right) \right|} \\ &\ge \frac{\lambda L_m}{\left| W\left(\frac{j}{(N_b - 1) N_b^{m+1}}\right) - W\left(\frac{j-1}{(N_b - 1) N_b^{m+1}}\right) \right|} \\ &= \frac{\lambda (N_b - 1) N_b L_{m+1}}{\left| W\left(\frac{j}{(N_b - 1) N_b^{m+1}}\right) - W\left(\frac{j-1}{(N_b - 1) N_b^{m+1}}\right) \right|} \\ &= \lambda (N_b - 1) N_b \tan \theta_{j-1,j,m+1} \\ &> (N_b - 1) \tan \theta_{j-1,j,m+1} \end{aligned} $$

since $\lambda N_b > 1$. Then, i. holds.

ii. One also has

$$ \theta_{j-1,j,m+1} < \arctan \frac{(N_b - 1) L_m}{h_{j-1,j,m}}, $$

where

$$ h_{j-1,j,m} = L_m^{2-D_W} \mathcal{O}(1) \quad \text{and} \quad C_{inf} \le \mathcal{O}(1) \le C_{sup}. $$

This yields

$$ \frac{(N_b - 1) L_m}{h_{j-1,j,m}} = L_m^{D_W - 1} \mathcal{O}(1) \quad \text{and} \quad (N_b - 1) C_{inf} \le \mathcal{O}(1) \le (N_b - 1) C_{sup}. $$

Consequently, $\theta_{j-1,j,m+1} \le L_m^{D_W-1}$, as claimed.

□
<figure><img src="image_12.png" /><figcaption>Fig. 9. An interior reentrant angle. Here, $N_b = 7$ and $\lambda = \frac{1}{2}$.</figcaption></figure>

are located within an epsilon distance from any edge; see, e.g., [38], [39], [41]. In the case of our fractal Weierstrass Curve $\Gamma_W$, which is, also, the limit of the sequence of (polygonal) prefractal graphs $(\Gamma_W)_{m \in \mathbb{N}}$, it is natural to envision the tubular neighborhood of $\Gamma_W$ as the limit (in the Hausdorff metric) of the (obviously convergent) sequence $(\mathcal{D}(\Gamma_m, \varepsilon_m^m))_{m \in \mathbb{N}}$ of $\varepsilon_m^m$-neighborhoods of $\Gamma_m$, where $\varepsilon = (\varepsilon_m^m)_{m \in \mathbb{N}}$ is a (suitable) infinitesimal – the cohomology infinitesimal – as introduced in Definition 3.1, on page 30 below. The cohomology infinitesimal is completely determined by the geometric characteristics of the fractal curve $\Gamma_W$ (or of the associated iterated fractal drum).

We note that, in a sense, the above description amounts to using a sequence of what we call *Weierstrass Iterated Fractal Drums* (in short, Weierstrass IFDs), by analogy with the *Relative Fractal Drums* (RFDs) introduced in [39], [40], [41]; for instance, in the case of the Cantor Staircase, in [39], Subsection 5.5.4, as well as in [40] and in [41]. In our present setting, the Weierstrass IFDs – i.e., the sets $\mathcal{D}(\Gamma_m, \varepsilon_m^m)$, for $m \in \mathbb{N}$ sufficiently large – contain the Weierstrass Curve $\Gamma_W$, and are sufficiently close to $\Gamma_W$, so that we can expect their Complex Dimensions to be closely related, as we will prove.

For this purpose, one thus requires fractal tube formulas for the sequence of prefractal graphs which converge to the Weierstrass Curve; i.e., here, the area of a two-sided $\varepsilon_m^m$-neighborhood of each prefractal approximation (with $m \in \mathbb{N}^*$ sufficiently large), which is expected to be of the following form, in the case of simple Complex Dimensions (for all $m \in \mathbb{N}$ sufficiently large, in our present case):

$$ \sum_{\omega \text{ Complex Dimension}} c_{\omega} \varepsilon^{2-\omega} , \quad c_{\omega} \in \mathbb{C}, \qquad (\star\star) $$
**Corollary 2.17 (Local Extrema of the Weierstrass Function (Coming from Property 2.10, on page 20)).**

i. The set of local maxima of the Weierstrass function on the interval $[0, 1]$ is given by

$$ \left\{ \left( \frac{(N_b - 1)k}{N_b^m}, W \left( \frac{(N_b - 1)k}{N_b^m} \right) \right) : 0 \le k \le N_b^m - 1, m \in \mathbb{N} \right\}, $$

and corresponds to the extreme vertices of the polygons at a given step $m$ (vertices connecting consecutive polygons).

ii. For odd values of $N_b$, the set of local minima of the Weierstrass function on the interval $[0, 1]$ is given by

$$ \left\{ \left( \frac{(N_b - 1)k + \frac{N_b - 1}{2}}{(N_b - 1)N_b^m}, W \left( \frac{(N_b - 1)k + \frac{N_b - 1}{2}}{(N_b - 1)N_b^m} \right) \right) : 0 \le k \le N_b^m - 1, m \in \mathbb{N} \right\}, $$

and corresponds to the bottom vertices of the polygons at a given step $m$.

**Property 2.18 (Existence of Reentrant Angles).**

i. The initial polygon $P_0$, admits **reentrant interior angles**, at a vertex $P_j$, with $0 < j \le N_b - 1$, in the sense that, with the right-hand rule, according to which angles are measured in a counter-clockwise direction $((P_jP_{j+1}), (P_jP_{j-1})) > \pi$, in the case when

$$ 0 < j \le \frac{N_b - 3}{4} \quad \text{or} \quad \frac{3N_b - 1}{4} \le j < N_b - 1 $$

(see Figure 7, on page 30), which does not occur for values of $N_b < 7$.

The number of reentrant angles is then equal to $2 \lfloor \frac{N_b - 3}{4} \rfloor$.

ii. At a given step $m \in \mathbb{N}^*$, with the above convention, a polygon $P_{m,k}$ admits reentrant interior angles in the sole cases when $N_b \ge 7$, at vertices $M_{k+j}$, $1 \le k \le N_b^m$, $0 < j \le N_b - 1$, as well as in the case when

$$ 0 < j \le \frac{N_b - 3}{4} \quad \text{or} \quad \frac{3N_b - 1}{4} \le j < N_b - 1 $$

The number of reentrant angles is then equal to $2 N_b^m \lfloor \frac{N_b - 3}{4} \rfloor$.
where, for any Complex Dimension $\omega$, $c_{\omega}$ is directly expressed in terms of the residue at $\omega$ of the $m^{th}$ effective prefractal tube zeta function $\tilde{\zeta}_{\omega}^{e}$ (or of the $m^{th}$ effective local distance zeta function $\zeta_{\omega}^{e}$).

More specifically, consistent with the corresponding results in [38], [39] and [41],

$$c_{\omega} = \operatorname{res} (\tilde{\zeta}_{\omega}^{e}, \omega) = \frac{1}{2-\omega} \operatorname{res} (\zeta_{\omega}^{e}, \omega).$$

We will proceed much as in [35], by the second author and E. P. J. Pearse, as well as in the later paper [36], by the same authors and S. Winter (see also [42], Section 10.3, or [43], Section 12.1), but, as will be explained further on, with several major differences. Note that these two papers were written prior to the development of the higher-dimensional theory of Complex Dimensions and fractal tube formulas, by the second author, G. Radunovic and D. Zubrinic, in the book [39] and in a series of accompanying papers by the same authors, including [38], [40] and [41].

The proper fractal zeta function to be used for this purpose, called the distance zeta function, was discovered by the second author in 2009, while the equivalent, and equally convenient, tube zeta function, depending on the problem at hand, was later introduced by the aforementioned authors in the above references. Both types of fractal zeta functions are connected via an explicit functional equation.

Consequently, once we have obtained the desired fractal tube formula for the Weierstrass IFD, we will be able to use extensions of the general results and methods of the higher-dimensional theory of Complex Dimensions in [38], [39], [40] and [41] in order to deduce the fractal zeta functions of the Weierstrass IFD: first, the so-called *effective prefractal tube zeta function*. We will then conclude from the expression of either fractal zeta function (since $D_{W} < 2$, they yield the same result here) the values of the (exact) Complex Dimensions of the Weierstrass IFD. For these Complex Dimensions, including the principal ones, in the terminology of [39] (i.e., those with real parts equal to the maximal real part $D_{W} < 2$), we will also be able to show that they are *actual* (and simple) Complex Dimensions of the Weierstrass IFD – that is, simple poles of the effective global tube zeta function.

As is explained in [14], the classical theory of Complex Dimensions (see, for instance, [39]) cannot be applied in the context of our fractal curve. Indeed, we were not, a priori, able to obtain the exact expression for the volume of the tubular neighborhood of the Weierstrass Curve, due to the extremely complicated geometric context. Building on the previous work of the second author and E. P. J. Pearse in the (less complicated) case of the Koch Curve [35], a possible method was to obtain an approximate expression for this tubular neighborhood. Recall that in the theory of Complex Dimensions, the imaginary parts of the Complex Dimensions aim at characterizing the (frequencies of the) oscillations of the fractal under study. Those oscillations are, also, connected to the evolution in scales – in real life (fractal-shaped living forms), as new details keep on appearing with characteristic spatial oscillations. In the aforementioned case of the Koch Curve (see [35]), the oscillations are obtained by means of Fourier series
<figure><img src="image_8.png" /><figcaption>Figure 7: An interior reentrant angle. Here, $N_b = 7$ and $\lambda = \frac{1}{2}$.</figcaption></figure>

Proof.

i. Due to the symmetry with respect to the vertical line $x = \frac{1}{2}$ (see Property 2.1, on page 9), one can restrict oneself to the vertices $P_j$, with $0 < j < \frac{N_b - 1}{2}$.

The initial polygon $P_0$, admits reentrant interior angles at a vertex $P_j$, with $j + 1 < \frac{N_b - 1}{2}$, in the case when

$$ ((y'y), (\overline{P_{j-1}P_j})) > ((y'y), (\overline{P_jP_{j+1}})) \quad (\spadesuit) \quad (\mathcal{R} 10) $$

Since

$$ P_j = (x_j, y_j) = \left( \frac{j}{N_b - 1}, \mathcal{W}\left(\frac{j}{N_b - 1}\right) \right) = \left( \frac{j}{N_b - 1}, \frac{1}{1 - \lambda} \cos\left(\frac{2\pi j}{N_b - 1}\right) \right), $$

one has

$$ \tan((y'y), \overline{(P_{j-1}P_j)}) = \frac{L_0}{|\mathcal{W}\left(\frac{j}{N_b - 1}\right) - \mathcal{W}\left(\frac{j-1}{N_b - 1}\right)|} $$

and

$$ \tan((y'y), \overline{(P_jP_{j+1}))} = \frac{L_0}{|\mathcal{W}\left(\frac{j+1}{N_b - 1}\right) - \mathcal{W}\left(\frac{j}{N_b - 1}\right)|}, $$

where $L_0 = \frac{1}{N_b - 1}$.

Therefore, condition $(\mathcal{R}10) - (\spadesuit)$ above corresponds to the case when

$$ |\mathcal{W}\left(\frac{j+1}{N_b - 1}\right) - \mathcal{W}\left(\frac{j}{N_b - 1}\right)| > |\mathcal{W}\left(\frac{j}{N_b - 1}\right) - \mathcal{W}\left(\frac{j-1}{N_b - 1}\right)|, $$
expansions of 1-periodic maps, where the 1-periodicity can be intuitively, understood in relation with the integers $m \in \mathbb{N}$ of the $m^{th}$ prefractal approximation.

Finally, we hereafter proceed in a much simpler and natural way. More precisely, building on the fact that the imaginary parts of the Complex Dimensions aim at characterizing the oscillations of the fractal under study are connected to the evolution in scales, we have introduced in [12] *cohomological vertex integers*, which not only enable us to obtain an exact series expansion, indexed by the Complex Codimensions, of the Weierstrass function $W$ itself, but, also, enable us to *label* each vertex of the prefractal graph at a given step of the prefractal approximation. This is just the analog of the method we have developed in our recent work on the Koch Curve [11], where the labels come from the fact that the triangular patterns which constitute the sequence of polyhedral neighborhoods are the images of the initial triangular pattern by the maps of the IFS associated with the Koch Curve – the counterpart of *the words* used in classical analysis on fractals (see, for instance, Definition 1.2.1, on page 12, in the book by Jun Kigami [23]). In our case, by introducing *cross-scales paths* (see Proposition 3.1 below, on page 33), we can, in some sense, *track* the evolution of vertices starting from the initial step of the prefractal approximation. In this light, our approach differs from the classical one.

Then, by using the aforementioned series expansion of the Weierstrass function $W$, we can obtain the exact expression for the elementary heights introduced in part iii. of Definition 2.5, on page 12. This enables us to obtain simpler estimates in the expression for the tubular volumes.

### 3.1. The tubular neighborhoods, and associated geometric characteristic numbers

**Notation 5 (Euclidean distance).** In the sequel, we denote by $d$ the Euclidean distance on $\mathbb{R}^2$.

Our results on fractal cohomology obtained in [12] have highlighted the part played by specific threshold values for the number $\varepsilon > 0$ at any step $m \in \mathbb{N}$ of the prefractal graph approximation; namely, the $m^{th}$ *cohomology infinitesimal* introduced in Definition 3.1, on page 30 just below.

**Definition 3.1** ($m^{th}$ *cohomology infinitesimal* [12] and intrinsic $m^{th}$ *cohomology infinitesimal*). From now on, given any $m \in \mathbb{N}$, we will call $m^{th}$ *cohomology infinitesimal* the number $\varepsilon_m^m > 0$ which is given by $\varepsilon_m^m = \frac{1}{N_b - 1} \frac{1}{N_b^m}$ (recall that $N_b > 1$), and coincides with the elementary horizontal length introduced in part i. of Definition 2.5, on page 12, i.e.,

$$L_m = \frac{1}{N_b - 1} \frac{1}{N_b^m}.$$
i.e.,

$$ \left| \cos\left(\frac{2\pi(j+1)}{N_b-1}\right) - \cos\left(\frac{2\pi j}{N_b-1}\right) \right| > \left| \cos\left(\frac{2\pi j}{N_b-1}\right) - \cos\left(\frac{2\pi(j-1)}{N_b-1}\right) \right|, $$

or, equivalently,

$$ \left| 2 \sin \frac{\pi}{N_b - 1} \sin \left( \frac{\pi (2j + 1)}{N_b - 1} \right) \right| > \left| 2 \sin \frac{\pi}{N_b - 1} \sin \left( \frac{\pi (2j - 1)}{N_b - 1} \right) \right|, $$

and thus happens if

$$ \left| \sin \left( \frac{\pi (2j + 1)}{N_b - 1} \right) \right| > \left| \sin \left( \frac{\pi (2j - 1)}{N_b - 1} \right) \right|. $$

Since

$$ 0 < \frac{\pi (2j - 1)}{N_b - 1} < \frac{\pi (2j + 1)}{N_b - 1} < \pi, $$

we conclude that condition (R10) – (♠), on page 30, occurs if

$$ 0 < \pi (2j - 1)N_b - 1 < \frac{\pi (2j + 1)}{N_b - 1} \le \frac{\pi}{2}, $$

i.e., if $0 < j \le \frac{N_b - 3}{4}$.

For vertices $P_j$, with $\frac{N_b + 1}{2} < j < N_b - 1$, the result is obtained thanks to the aforementioned symmetry. The initial polygon $P_0$, admits **reentrant interior angles** at a vertex $P_j$ in the case when $\frac{3N_b - 1}{4} \le j < N_b - 1$.

ii. The result is obtained by strong induction on the integer $m$. We restrict ourselves to the values $N_b \ge 7$, and consider $j$ in $\{0, \dots, \lfloor \frac{N_b - 3}{4} \rfloor\}$.

We claim that the result is satisfied at the initial step $m = 1$. Indeed, as was already encountered in the proof of Property 2.10, on page 20, for any integer $k$ in $\{0, \dots, N_b - 1\}$, we have that

$$ \left| W\left(\frac{k(N_b - 1) + j + 1}{(N_b - 1)N_b}\right) - W\left(\frac{j}{(N_b - 1)N_b}\right) \right| = \left| (1 + \lambda) \left\{ W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j}{N_b-1}\right) \right\} \right| $$

and

$$ \left| W\left(\frac{k(N_b - 1) + j}{(N_b - 1)N_b}\right) - W\left(\frac{j-1}{(N_b - 1)N_b}\right) \right| = \left| (1 + \lambda) \left\{ W\left(\frac{j}{N_b-1}\right) - W\left(\frac{j-1}{N_b-1}\right) \right\} \right|. $$
Observe that, clearly, $\varepsilon_m$ itself – and not just $\varepsilon_m^m$ – depends on $m$; hence, we should really write $\varepsilon_m^m = (\varepsilon_m)^m$, for all $m \in \mathbb{N}$.

In addition, since $N_b > 1$, $\varepsilon_m^m$ satisfies the following asymptotic behavior,

$$ \varepsilon_m^m \to 0, \text{ as } m \to \infty, $$

which, naturally, results from the fact that the larger $m$, the smaller $\varepsilon_m^m$. It is for this reason that we call $\varepsilon_m^m$ – or rather, the *infinitesimal sequence* $(\varepsilon_m^m)_{m=0}^\infty$ of positive numbers tending to zero as $m \to \infty$, with $\varepsilon_m^m = (\varepsilon_m)^m$, for each $m \in \mathbb{N}$ – an *infinitesimal*. Note that this $m^{th}$ cohomology infinitesimal is the one naturally associated to the scaling relation of Property 2.8, on page 18.

In the sequel, it is also useful to keep in mind that the sequence of positive numbers $(\varepsilon_m)_{m=0}^\infty$ itself satisfies

$$ \varepsilon_m \sim \frac{1}{N_b}, \text{ as } m \to \infty; $$

i.e., $\varepsilon_m \to \frac{1}{N_b}$, as $m \to \infty$. In particular, $\varepsilon_m \not\to 0$, as $m \to \infty$, but, instead, $\varepsilon_m$ tends to a strictly positive and finite limit.

We also introduce, given any $m \in \mathbb{N}$, the $m^{th}$ intrinsic cohomology infinitesimal, denoted by $\varepsilon^m > 0$, such that

$$ \varepsilon^m = \frac{1}{N_b^m}, $$

where

$$ \varepsilon = \frac{1}{N_b}. $$

We call $\varepsilon$ the intrinsic scale, or intrinsic subdivision scale.

Note that

$$ \varepsilon_m^m = \frac{\varepsilon^m}{N_b - 1} = L_m $$

and that, up to a multiplicative constant (namely, $\frac{1}{N_b - 1}$), the $m^{th}$ intrinsic cohomology infinitesimal $\varepsilon^m$ is asymptotic (when $m$ tends to $\infty$) to the $m^{th}$ cohomology infinitesimal $\varepsilon_m^m$.

**Remark 3.1 (Addressing numerical estimates).** From a practical point of view, an important question is the value of the ratio

$$ \frac{\text{Cohomology infinitesimal}}{\text{Maximal height}} = \frac{\varepsilon_m^m}{h_m}; $$
Thus,

$$ 
\begin{aligned}
\frac{\tan \theta_k(N_b-1)+j-1, k(N_b-1)+j, 1}{\tan \theta_k(N_b-1)+j, k(N_b-1)+j+1, 1} &= \frac{\left|W\left(\frac{k(N_b-1)+j+1}{N_b-1}\right) - W\left(\frac{k(N_b-1)+j}{N_b-1}\right)\right|}{\left|W\left(\frac{k(N_b-1)+j}{N_b-1}\right) - W\left(\frac{k(N_b-1)+j-1}{N_b-1}\right)\right|} \\
&= \frac{\left|W\left(\frac{j+1}{N_b-1}\right) - W\left(\frac{j-1}{N_b-1}\right)\right|}{\left|W\left(\frac{j}{N_b-1}\right) - W\left(\frac{j-1}{N_b-1}\right)\right|} > 1,
\end{aligned}
$$

which implies that

$$ \theta_k(N_b-1)+j-1, k(N_b-1)+j, 1 > \theta_k(N_b-1)+j, k(N_b-1)+j+1, 1 $$

and yields the existence of an interior reentrant angle at the vertex

$$ \left( \frac{k(N_b - 1) + j}{N_b - 1}, W\left( \frac{k(N_b - 1) + j}{N_b - 1} \right) \right). $$

Let us now assume that, up to a given step $m \ge 1$, there is a reentrant interior angle at any vertex

$$ \left( \frac{k(N_b - 1) + j}{N_b - 1} N_b^{m-1}, W\left( \frac{k(N_b - 1) + j}{N_b - 1} N_b^{m-1} \right) \right) , \quad \text{with } 0 \le k \le N_b^{m-1} - 1. $$

We then want to prove that there is a reentrant interior angle at any vertex

$$ \left( \frac{k(N_b - 1) + j}{N_b - 1} N_b^m, W\left( \frac{k(N_b - 1) + j}{N_b - 1} N_b^m \right) \right) , \quad \text{with } 0 \le k \le N_b^m - 1. $$

As was the case in the proof of Property 2.10 (page 20), in order to be able to use the induction hypothesis, we express any integer $k$ in $\{0, \dots, N_b^m - 1\}$ in the following form:

$$ k = \tilde{k} + i N_b^{m-1} , \quad 0 \le \tilde{k} \le N_b^{m-1} - 1 , \quad 0 \le i \le N_b - 1. \qquad (\mathcal{R} 11) $$

Thus,

$$ W\left(\frac{k(N_b - 1) + j + 1}{N_b - 1} N_b^m\right) - W\left(\frac{j}{N_b - 1} N_b^m\right) = \lambda \left( W\left(\frac{\tilde{k}(N_b - 1) + j + 1}{N_b - 1} N_b^{m-1}\right) - W\left(\frac{\tilde{k}(N_b - 1) + j}{N_b - 1} N_b^{m-1}\right) \right) \\ -2 \sin\left(\frac{\pi}{N_b - 1} N_b^{m-1}\right) \sin\left(\frac{2(N_b - 1)\tilde{k} + 2j + 1}{N_b - 1} N_b^{m-1}\right), \qquad (\mathcal{R} 12) $$

and

$$ W\left(\frac{k(N_b - 1) + j}{N_b - 1} N_b^m\right) - W\left(\frac{j-1}{N_b - 1} N_b^m\right) = \lambda \left( W\left(\frac{\tilde{k}(N_b - 1) + j}{N_b - 1} N_b^{m-1}\right) - W\left(\frac{\tilde{k}(N_b - 1) + j - 1}{N_b - 1} N_b^{m-1}\right) \right) \\ -2 \sin\left(\frac{\pi}{N_b - 1} N_b^{m-1}\right) \sin\left(\frac{2(N_b - 1)\tilde{k} + 2j - 1}{N_b - 1} N_b^{m-1}\right). \qquad (\mathcal{R} 13) $$
see relation (5), on page 15.

Thanks to the estimates given in Remark 2.3, on page 23, we have that

$$0 < C_{inf} \varepsilon_m^m (1-D_W) \le \frac{\varepsilon_m^m}{h_m} \le C_{sup} \varepsilon_m^m (1-D_W).$$

Given $q \in \mathbb{N}^*$, we then have

$$\frac{1}{10^q} C_{inf} \le \frac{\varepsilon_m^m}{h_m} \le \frac{1}{10^q} C_{sup}$$

when

$$\frac{C_{inf}}{10^q} \le e^{(1-D_W) \ln L_m} \le \frac{C_{sup}}{10^q},$$

or, equivalently, when

$$-\frac{1}{\ln N_b} \ln \left( (N_b - 1) \left( \frac{C_{sup}}{10^q} \right)^{\frac{1}{1-D_W}} \right) \le m \le -\frac{1}{\ln N_b} \ln \left( (N_b - 1) \left( \frac{C_{inf}}{10^q} \right)^{\frac{1}{1-D_W}} \right).$$

Numerical values for $N_b = 3$ and $\lambda = \frac{1}{2}$ yield:

i. For $q = 1$: $2 \le m \le 3$.

ii. For $q = 2$: $7 \le m \le 9$.

iii. For $q = 3$: $13 \le m \le 15$.

Hence, when $m$ increases, the ratio $\frac{\varepsilon_m^m}{h_m}$ decreases, and tends to 0. This numerical – but very practical and explicit argument – also applies to our forthcoming neighborhoods, of width equal to the cohomology infinitesimal.

**Definition 3.2 (Cohomological vertex integers [13]).** Given $m \in \mathbb{N}$, and a vertex $M_{j,m} = M_{(N_b-1)k'+k'',m} \in V_m$, of abscissa $((N_b-1)k'+k'') \varepsilon_m^m$, where $0 \le k' \le N_b^m - 1$ and $0 \le k'' \le N_b - 1$, we introduce the cohomological vertex integer $\ell_{j,m}$ associated to the vertex $M_{j,m}$ (which is also the $(k'')^{th}$ vertex of the polygon $P_{m,k'}$; see part iii. of Property 2.7, on page 15), as

$$\ell_{j,m} = \ell_{k',k'',m} = (N_b - 1) k' + k''. \qquad (11)$$

Depending on the context; that is,

i. when the cohomological vertex integer enables one to locate the vertex $M_{j,m}$.

ii. When it is used in a more general framework, i.e., in order to describe the generators of cohomology groups (see [10]);
In light of Property 2.10, on page 20, given such an integer $k$ - and hence also, $\tilde{k}$ and $j$ – and since

$$0 \le j \le \left\lfloor \frac{N_b - 3}{4} \right\rfloor \le \frac{N_b - 1}{2},$$

the only configuration to be considered corresponds to the case when

$$\theta_{\tilde{k}}(N_b-1)+j-1, \tilde{k}(N_b-1)+j, m-1 > \theta_{\tilde{k}}(N_b-1)+j, \tilde{k}(N_b-1)+j+1, m-1$$

and

$$W\left(\frac{\tilde{k}(N_b - 1) + j - 1}{(N_b - 1) N_b^{m-1}}\right) - W\left(\frac{j}{(N_b - 1) N_b^{m-1}}\right) > 0, \quad W\left(\frac{\tilde{k}(N_b - 1) + j}{(N_b - 1) N_b^{m-1}}\right) - W\left(\frac{\tilde{k}(N_b - 1) + j + 1}{(N_b - 1) N_b^{m-1}}\right) > 0.$$

Then,

$$\tan \theta_{\tilde{k}}(N_b-1)+j-1, \tilde{k}(N_b-1)+j, m-1 > \tan \theta_{\tilde{k}}(N_b-1)+j, \tilde{k}(N_b-1)+j+1, m-1;$$

i.e.,

$$\left| W\left(\frac{\tilde{k}(N_b - 1) + j}{(N_b - 1) N_b^{m-1}}\right) - W\left(\frac{\tilde{k}(N_b - 1) + j - 1}{(N_b - 1) N_b^{m-1}}\right) \right| > \left| W\left(\frac{\tilde{k}(N_b - 1) + j + 1}{(N_b - 1) N_b^{m-1}}\right) - W\left(\frac{\tilde{k}(N_b - 1) + j}{(N_b - 1) N_b^{m-1}}\right) \right|,$$

which yields

$$\left| W\left(\frac{\tilde{k}(N_b - 1) + j + 1}{(N_b - 1) N_b^{m-1}}\right) - W\left(\frac{\tilde{k}(N_b - 1) + j}{(N_b - 1) N_b^{m-1}}\right) \right| > \left| W\left(\frac{\tilde{k}(N_b - 1) + j}{(N_b - 1) N_b^{m-1}}\right) - W\left(\frac{\tilde{k}(N_b - 1) + j - 1}{(N_b - 1) N_b^{m-1}}\right) \right|,$$

or, equivalently,

$$W\left(\frac{\tilde{k}(N_b - 1) + j}{(N_b - 1) N_b^{m-1}}\right) - W\left(\frac{\tilde{k}(N_b - 1) + j + 1}{(N_b - 1) N_b^{m-1}}\right) > W\left(\frac{\tilde{k}(N_b - 1) + j - 1}{(N_b - 1) N_b^{m-1}}\right) - W\left(\frac{\tilde{k}(N_b - 1) + j}{(N_b - 1) N_b^{m-1}}\right).$$

The strong induction hypothesis, which ensures the existence of a reentrant interior angle at the vertex

$$\left( \frac{(N_b - 1) \tilde{k} + j}{(N_b - 1) N_b^{m-2}}, W\left( \frac{(N_b - 1) \tilde{k} + j}{(N_b - 1) N_b^{m-2}} \right) \right),$$

requires, in conjunction with

$$W\left(\frac{\tilde{k}(N_b - 1) + j}{(N_b - 1) N_b^{m-2}}\right) - W\left(\frac{\tilde{k}(N_b - 1) + j + 1}{(N_b - 1) N_b^{m-2}}\right) > W\left(\frac{\tilde{k}(N_b - 1) + j - 1}{(N_b - 1) N_b^{m-2}}\right) - W\left(\frac{\tilde{k}(N_b - 1) + j}{(N_b - 1) N_b^{m-2}}\right),$$
we will use the best suited notation between $\ell_{j,m}$, in case i., or $\ell_{k',k'',m}$, in case ii.

**Proposition 3.1 (Cross-scales paths, and associated sequence of vertex integers).**
Given $m \in \mathbb{N}$, $0 \le j \le \#V_m - 1$ and a vertex $M_{j,m} = M_{(N_b-1)k'+k'',m}$ in $V_m$, with $0 \le k' \le N_b^m - 1$ and $0 \le k'' \le N_b - 1$, we introduce the cross-scales path $\mathcal{P}ath (P_{k''}, M_{j,m})$, where $P_{k''}$ is the $(k'')^{th}$ fixed point of the map $T_{k''}$ (see Proposition 2.2, on page 10, along with Definition 2.2, on page 11), as the ordered set $(M_{j_{k,m},k})_{0 \le k \le m}$ such that:

i. For $0 \le k \le m$, each vertex $M_{j_{k,m},k}$ strictly belongs to $V_k$, i.e., it is in the $k^{th}$ pre-fractal approximation $\Gamma_{W_k}$, and not in $\Gamma_{W_{k+1}}$.

ii. For $1 \le k \le m$, each vertex $M_{j_{k,m},k} = M_{(N_b-1)k'+k'',k}$, with $0 \le k'_{k,m} \le N_b^k - 1$, is the image of the point $M_{j_{k-1,m},k-1}$ under the map $T_i$ (see again Proposition 2.2, on page 10), where $i \in \{0, \dots, N_b - 1\}$ is the smallest admissible value. We thus also have that

$$M_{j_{k-1,m},k-1} = \left( x_{(N_b-1)(k'_{k,m}-i(N_b-1)N_b^{k-1})+k'',k-1}, \quad \mathcal{W}\left(x_{(N_b-1)(k'_{k,m}-i(N_b-1)N_b^{k-1})+k'',k-1}\right) \right).$$

This latter point is also the $(k'')^{th}$ vertex of the polygon $k'_{k,m} - i(N_b - 1)N_b^{k-1}$ (see part iii. of Property 2.7, on page 15).

The sequence of vertex integers associated with the cross-scales path $\mathcal{P}ath (P_{k''}, M_{j,m})$ (or, in short, and equivalently, also referred to as the sequence of vertex integers associated with $M_{j,m}$) is the sequence $(\ell_{j_{k,m},k})_{0 \le k \le m}$, where, for $0 \le k \le m$, $\ell_{j_{k,m},k}$ is the cohomological vertex integer associated with the vertex $M_{j_{k,m},k}$ (see Definition 3.2, on page 32).

**Proof.** We simply use the results of Property 2.11, on page 21. □

**Definition 3.3 (Complexified Weierstrass function).** We introduce the Complexified Weierstrass function $W_{comp}$, defined, for any real number $x$, by

$$W_{comp}(x) = \sum_{n=0}^{\infty} \lambda^n e^{2i \pi N_b^n x}.$$

Clearly, $W_{comp}$ is also a continuous and 1-periodic function on $\mathbb{R}$ and $W$ is the real part of $W_{comp}$.

**Theorem 3.2 (Complex dimensions series expansion of the complexified Weierstrass function $W_{comp}$ [12], and of the Weierstrass function $W$).** For any sufficiently large positive
that

$$ \sin\left(\frac{\pi(2\tilde{k}(N_b-1)+2j+1)}{(N_b-1)N_b^{m-2}}\right) > \sin\left(\frac{\pi(2\tilde{k}(N_b-1)+2j-1)}{(N_b-1)N_b^{m-2}}\right), $$

which corresponds to

$$ 0 < \frac{\pi(2\tilde{k}(N_b-1)+2j+1)}{(N_b-1)N_b^{m-2}} < \frac{\pi(2\tilde{k}(N_b-1)+2j-1)}{(N_b-1)N_b^{m-2}} \le \frac{\pi}{2} $$

and, as a matter of fact, ensures that

$$ 0 < \frac{\pi(2\tilde{k}(N_b-1)+2j+1)}{(N_b-1)N_b^{m-1}} < \frac{\pi(2\tilde{k}(N_b-1)+2j-1)}{(N_b-1)N_b^{m-21}} \le \frac{\pi}{2N_b} < \frac{\pi}{2}. $$

One then has the following inequality:

$$ \lambda \left( W\left( \frac{\tilde{k}(N_b-1)+j}{(N_b-1)N_b^{m-1}} \right) - W\left( \frac{\tilde{k}(N_b-1)+j+1}{(N_b-1)N_b^{m-1}} \right) \right) + 2 \sin\left( \frac{\pi}{(N_b-1)N_b^{m-1}} \right) \sin\left( \frac{\pi(2\tilde{k}(N_b-1)+2j+1)}{(N_b-1)N_b^{m-1}} \right) \\ > \lambda \left( W\left( \frac{\tilde{k}(N_b-1)+j-1}{(N_b-1)N_b^{m-1}} \right) - W\left( \frac{\tilde{k}(N_b-1)+j}{(N_b-1)N_b^{m-1}} \right) \right) + 2 \sin\left( \frac{\pi}{(N_b-1)N_b^{m-1}} \right) \sin\left( \frac{\pi(2\tilde{k}(N_b-1)+2j-1)}{(N_b-1)N_b^{m-1}} \right). $$

Hence,

$$ \tan \theta_{\tilde{k}(N_b-1)+j-1, \tilde{k}(N_b-1)+j, m} $$

||

$$ \begin{aligned} & \left| \frac{L_m}{W\left(\frac{\tilde{k}(N_b-1)+j}{(N_b-1)N_b^m}\right) - W\left(\frac{\tilde{k}(N_b-1)+j-1}{(N_b-1)N_b^m}\right)} \right| \\ = & \left| \lambda \left( W\left(\frac{\tilde{k}(N_b-1)+j}{(N_b-1)N_b^m}\right) - W\left(\frac{\tilde{k}(N_b-1)+j-1}{(N_b-1)N_b^m}\right) \right) - 2 \sin\left(\frac{\pi}{(N_b-1)N_b^{m-1}}\right) \sin\left(\frac{\pi(2\tilde{k}(N_b-1)+2j-1)}{(N_b-1)N_b^{m-1}}\right) \right| \\ = & \frac{L_m}{\lambda \left( W\left(\frac{\tilde{k}(N_b-1)+j-1}{(N_b-1)N_b^m}\right) - W\left(\frac{j}{(N_b-1)N_b^m}\right) \right) + 2 \sin\left(\frac{\pi}{(N_b-1)N_b^{m-1}}\right) \sin\left(\frac{\pi(2\tilde{k}(N_b-1)+2j-1)}{(N_b-1)N_b^{m-1}}\right)} \\ > & \frac{L_m}{\lambda \left( W\left(\frac{\tilde{k}(N_b-1)+j}{(N_b-1)N_b^m}\right) - W\left(\frac{\tilde{k}(N_b-1)+j+1}{(N_b-1)N_b^m}\right) \right) + 2 \sin\left(\frac{\pi}{(N_b-1)N_b^{m-1}}\right) \sin\left(\frac{\pi(2\tilde{k}(N_b-1)+2j+1)}{(N_b-1)N_b^{m-1}}\right)} \\ = & \tan \theta_{\tilde{k}(N_b-1)+j, \tilde{k}(N_b-1)+j+1, m}, \end{aligned} $$

which yields the expected result. Namely,

$$ \theta_{\tilde{k}(N_b-1)+j-1, \tilde{k}(N_b-1)+j, m} > \theta_{\tilde{k}(N_b-1)+j, \tilde{k}(N_b-1)+j+1, m}; $$

i.e., the presence of a reentrant angle at the $j^{th}$ vertex of the polygon $P_{m,k}$.

The result in the remaining case $\frac{3N_b-1}{4} \le j < N_b-1$ can be obtained in an entirely similar way. It corresponds to the cases when
integer $m$ and any $j$ in $\{0, \dots, \#V_m - 1\}$, we have the following exact expansion, indexed by the Complex Codimensions $k$ ($D_W - 2) + ik \ell_{j_{k,m},k} \mathbf{p}$, with $0 \le k \le m$,

$$ 
\begin{aligned}
W_{\text{comp}}(x_{j,m}) &= W_{\text{comp}}\left(\frac{j \varepsilon^m}{N_b - 1}\right) \\
&= \varepsilon^m (2 - D_W) W_{\text{comp}}\left(\frac{j}{N_b - 1}\right) + \sum_{k=0}^{m-1} c_{k,j,m} \varepsilon^k (2 - D_W) \varepsilon^{i \ell_{j_{k,m},k}} \mathbf{p} \quad (12) \\
&= \sum_{k=0}^{m} c_{k,j,m} \varepsilon^k (2 - D_W) \varepsilon^{i \ell_{j_{k,m},k}} \mathbf{p},
\end{aligned}
$$

where, for $0 \le k \le m$, $\varepsilon^k$ is the $k^{th}$ intrinsic cohomology infinitesimal, introduced in Definition 3.1, on page 30, with $\mathbf{p} = \frac{2\pi}{\ln N_b}$ denoting the oscillatory period of the (complexified) Weierstrass Curve and where:

i. $\ell_{j_{k,m},k} \in \mathbb{N}$ is the cohomological vertex integer associated with the vertex $M_{j_{k,m},k}$ (see Definition 3.2, on page 32);

ii. $c_{m,j,m} = W_{\text{comp}}\left(\frac{j}{N_b - 1}\right)$ and, for $0 \le k \le m - 1$, $c_{k,j,m} \in \mathbb{C}$ is given by

$$ c_{k,j,m} = \exp \left( \frac{2i\pi}{N_b - 1} j \varepsilon^{m-k} \right) . \quad (\diamond\diamond) \quad (13) $$

For $0 \le k \le m$, the coefficient $c_{k,j,m}$ will also be referred to as the $k^{th}$ Weierstrass coefficient associated with the vertex $M_{j_{k,m},k} \in V_k$.

For any $m \in \mathbb{N}$, the complex numbers $\{c_{0,j,m+1}, \dots, c_{m+1,j,m+1}\}$ satisfy the following recurrence relations:

$$ c_{m+1,j,m+1} = W\left(\frac{j}{N_b - 1}\right) = c_{m,j,m} \quad (14) $$

and

$$ \forall k \in \{1, \dots, m\} : \quad c_{k,j,m+1} = c_{k-1,j,m}. \quad (15) $$

In addition, since relation (12) is valid for any $m \in \mathbb{N}^*$ (and since, clearly, relation (13) implies that the coefficients $c_{k,j,m}$ are nonzero for $0 \le k \le m$), we deduce that the associated Complex Dimensions (i.e., in fact, the Complex Dimensions associated with the complexified Weierstrass function $W_{\text{comp}}$) are, still for all $m \in \mathbb{N}^*$ sufficiently large,

$$ D_W - k(2 - D_W) + i \ell_{j_{k,m},k} \mathbf{p}, $$
$$ \theta_k(N_b-1)+j-1, \tilde{k}(N_b-1)+j, mc < \theta_k(N_b-1)+j, \tilde{k}(N_b-1)+j+1, m $$

and

$$ W\left(\frac{k(N_b - 1) + j - 1}{(N_b - 1) N_b^m}\right) - W\left(\frac{k(N_b - 1) + j}{(N_b - 1) N_b^m}\right) < 0 \quad , \quad W\left(\frac{k(N_b - 1) + j}{(N_b - 1) N_b^m}\right) - W\left(\frac{k(N_b - 1) + j + 1}{(N_b - 1) N_b^m}\right) < 0 $$

Therefore, the shape of the initial polygon $P_0$ governs the shape of any polygon $P_{m,k}$, $0 \le k \le N_b^m$, which, if $N_b \ge 7$, admits reentrant interior angles at vertices $M_{(N_b-1)k+j}$, $0 \le k \le N_b^m - 1$, $0 < j \le N_b - 1$, in the case when

$$ 0 < j \le \frac{N_b - 3}{4} \quad \text{or} \quad \frac{3N_b - 1}{4} \le j < N_b - 1 $$

This concludes the proof of Property 2.18 given on page 29.

**Definition 2.6 (Self-Shape Similarity of the Weierstrass Curve).**

We will say that the Weierstrass Curve – as the two-dimensional Hausdorff and uniform limit curve of a sequence of polygonal prefraçals, which satisfy Property 2.10, on page 20 and Property 2.18, on page 29 – has self-shape similarity, in the sense that the shape of the initial polygon $P_0$ governs the shape of all the polygons $P_{m,k}$, with $0 \le k \le N_b^m$, at any step $m$ of the prefraçal approximation process. This self-shape similarity property is apparent in Figure 1, on page 11, Figure 2, on page 12, and Figure 3, on page 13. As for the existence of reentrant angles, it can be observed on the first two graphs of Figure 3, on page 13, in the case when $N_b = 7$.

## 3 Iterated Fractal Drums and Tubular Neighborhoods

In the case of classical fractals, and when the associated geometry allows it, the values of the Complex Dimensions are obtained by studying the oscillations of a small neighborhood of the boundary, i.e., of a tubular neighborhood of the fractal, where points are located within an epsilon distance from any edge; see, e.g., [LRŽ17a], [LRŽ17b], [LRŽ18]. In the case of our fractal Weierstrass Curve $\Gamma_W$, which is, also, the limit of the sequence of (polygonal) prefraçal graphs $(\Gamma_W)_{m \in \mathbb{N}}$, it is natural – and consistent with the result of Property 3.9, on page 59 below – to envision the tubular neighborhood of $\Gamma_W$ as the limit of the (obviously convergent) sequence $(\mathcal{D}(\Gamma_{W_m}, \varepsilon_m^m))_{m \in \mathbb{N}}$ of $\varepsilon_m^m$-neighborhoods of $\Gamma_{W_m}$, where $\varepsilon = (\varepsilon_m^m)_{m \in \mathbb{N}}$ is a (suitable) infinitesimal – the cohomology infinitesimal – as introduced in Definition 3.1, on page 37 below. The cohomology infinitesimal is completely determined by the geometric characteristics of the fractal curve $\Gamma_W$ (or of the associated iterated fractal drum).

We note that, in a sense, the above description amounts to using a sequence of what we call *Weierstrass Iterated Fractal Drums* (in short, Weierstrass IFDs), by analogy with the *Relative Fractal Drums* (RFDs), for instance, in the case of the Cantor Staircase, in [LRŽ17b], Section 5.5.4, as well as in [LRŽ17c] and in [LRŽ18]. In our present setting, the Weierstrass IFDs – i.e., the sets $\mathcal{D}(\Gamma_{W_m}, \varepsilon_m^m)$, for $m \in \mathbb{N}$ sufficiently large – contain the Weierstrass Curve $\Gamma_W$, and are sufficiently close to $\Gamma_W$, so that we can expect their Complex Dimensions to be the same.
where $0 \le k \le m$ and $\ell_{j_k, m, k} \in \mathbb{N}$ is the cohomological vertex integer associated with the vertex $M_{j_k, m, k}$ (see Definition 3.2, on page 32). Those Complex Dimensions are all exact and simple.

This immediately ensures, for the Weierstrass function $W$ (i.e., the real part of the Complexified Weierstrass function $W_{comp}$), that, for any strictly positive integer $m$ sufficiently large and for any $j$ in $\{0, \dots, \#V_m - 1\}$,

$$ 
\begin{aligned}
W(x_{j,m}) &= \varepsilon^{m(2-D_W)} W_{\text{comp}}\left(\frac{j}{N_b-1}\right) + \sum_{k=0}^{m-1} \varepsilon^{k(2-D_W)} \operatorname{Re}\left(c_{k,j,m} \varepsilon_k^{i\ell_{j_k,m,k} \mathbf{P}}\right) \\
&= \varepsilon^{m(2-D_W)} W_{\text{comp}}\left(\frac{j}{N_b-1}\right) \\
&\quad + \frac{1}{2} \sum_{k=0}^{m-1} \varepsilon^{k(2-D_W)} \left(c_{k,j,m} \varepsilon^{i\ell_{j_k,m,k} \mathbf{P}} + \overline{c_{k,j,m}} \varepsilon^{-i\ell_{j_k,m,k} \mathbf{P}}\right) \quad (16) \\
&= \frac{1}{2} \sum_{k=0}^{m} \varepsilon^{k(2-D_W)} \left(c_{k,j,m} \varepsilon^{i\ell_{j_k,m,k} \mathbf{P}} + \overline{c_{k,j,m}} \varepsilon^{-i\ell_{j_k,m,k} \mathbf{P}}\right),
\end{aligned}
$$

where $\bar{z}$ denotes the complex conjugate of $z \in \mathbb{C}$.

More generally, for any strictly positive integer $m$ sufficiently large and for any integer $j$ in $\{0, \dots, \#V_m - 1\}$,

$$ W_{\text{comp}}(j \varepsilon^m) = \sum_{k=0}^{\infty} \varepsilon^k (2-D_W) c_{k,j,m} \varepsilon^k (2-D_W) \varepsilon^{i \ell_{j_k,m,k} \mathbf{P}}, \quad (17) $$

where, for all $k \in \mathbb{N}$,

$$ c_{k,j,m} = \varepsilon^{2i \pi N_b^k} j \varepsilon^m. \quad (18) $$

We also note that, if a vertex $M_{j,m} = M_{j',m+m'}$ is in $V_m \cap V_{m+m'}$, for $m' \in \mathbb{N}$, we of course have that, for $0 \le k \le m$,

$$ c_{k,j,m} = c_{k,j',m+m'}, \quad (19) $$

along with

$$ \varepsilon^{i \ell_{j_k, m, k}} = \varepsilon^{i \ell_{j_k, m+m'', k}}. \quad (20) $$

For $m+1 \le k \le m+m'$, we have that

$$ c_{k,j,m} = c_{k,j',m+m'} = 0. \quad (21) $$

In addition, we have that, for any $m' \in \mathbb{N}$,

$$ c_{k,j,m+m'} = \lambda^{m'} c_{k,j,m} = \varepsilon^{m'} (2-D_W) c_{k,j,m}. \quad (22) $$
For this purpose, one thus requires fractal tube formulas for the sequence of prefractal graphs which converge to the Weierstrass Curve; i.e., here, the area of a two-sided $\epsilon_m^m$-neighborhood of each prefractal approximation (with $m \in \mathbb{N}^*$ sufficiently large), which is expected to be of the following form, in the case of simple Complex Dimensions:

$$ \sum_{\omega \text{ Complex Dimension}} c_{\omega} (\epsilon_m^m)^{2-\omega}, \quad c_{\omega} \in \mathbb{C}, \quad (\star\star) $$

where, for any Complex Dimension $\omega$, $c_\omega$ is directly expressed in terms of the residue at $\omega$ of the effective tube zeta function $\tilde{\zeta}_\omega^e$ (or of the effective distance zeta function $\zeta_\omega^e$).

More specifically, consistent with the corresponding results in [LRŽ17a], [LRŽ17b] and [LRŽ18],

$$ c_{\omega} = \mathrm{res} \left( \tilde{\zeta}_{\omega}^{e}, \omega \right) = \frac{1}{2-\omega} \mathrm{res} \left( \zeta_{\omega}^{e}, \omega \right). $$

We shall proceed as in [LP06], by the second author and E. P. J. Pearse, as well as in the later paper [LPW11], by the same authors and S. Winter (see also [LvF00], §10.3, or [LvF06], §12.1). Note that these two papers were written prior to the development of the higher-dimensional theory of Complex Dimensions and fractal tube formulas, by the second author, G. Radunovic and D. Zubrinic, in the book [LRŽ17b] and in a series of accompanying papers by the same authors, including [LRŽ17a], [LRŽ18].

The proper fractal zeta function to be used for this purpose, called the distance zeta function, was discovered by the second author in 2009, while the equivalent, and equally convenient, tube zeta function, depending on the problem at hand, was later introduced by the aforementioned authors in the above references. Both types of fractal zeta functions are connected via an explicit functional equation.

Consequently, once we have obtained the desired fractal tube formula for the Weierstrass IFD, we will be able to use extensions of the general results and methods of the higher-dimensional theory of Complex Dimensions in [LRŽ17a], [LRŽ17b] and [LRŽ18] in order to deduce the fractal zeta functions of the Weierstrass IFD: first, the so-called *effective tube zeta function* and then, via the aforementioned functional equation connecting those two zeta functions, the *effective distance zeta function*. We will then conclude from the expression of either fractal zeta function (since $D_W < 2$, they yield the same result here) the values of the possible Complex Dimensions of the Weierstrass IFD. For many of those Complex Dimensions, including the principal ones, in the terminology of [LRŽ17b] (i.e., those with real parts equal to the maximal real part $D_W < 2$), we will also be able to determine that they are *actual* (and simple) Complex Dimensions of the Weierstrass IFD – that is, simple poles of the tube zeta function, or, equivalently, of the distance zeta function.

We note that the only possible exceptions to the latter statement would be the potential Complex Dimensions with real part equal to 1 (except for 1 itself), some (or all) of which could have a vanishing residue; further theoretical or numerical work will be needed in order to deal with this last remaining issue.

### Notation 8 (Euclidean Distance).

In the sequel, we denote by $d$ the Euclidean distance on $\mathbb{R}^2$.
**Corollary 3.3.** For any sufficiently large positive integer $m$ and any $j$ in $\{1, \dots, \#V_m - 1\}$, we have the following exact expansion, indexed by the Complex Codimensions $k$ ($D_W - 2$) + $i \ell_{k,j,m} \mathbf{p}$, with $0 \le k \le m$,

$$h_{j-1,m} = \sum_{k=0}^{m} d_{k,j,m} \varepsilon^{k} (2-D_{W}) \varepsilon^{i \ell_{k,j,m}} \mathbf{p}, \quad (23)$$

where, for $0 \le k \le m$, $\varepsilon^k$ is the $k^{th}$ intrinsic cohomology infinitesimal, introduced in Definition 3.1, on page 30, and

i. in the case when $h_{j-1,m} \ge 0$,

$$d_{k,j,m} = c_{k,j,m} - c_{k,j-1,m} \varepsilon^{i (\ell_{j,k-1,m,k} - \ell_{j,k,m,k})} \mathbf{p}$$

and

$$\ell_{k,j,m} = \ell_{j,k,m,k};$$

ii. In the case when $h_{j-1,m} \le 0$,

$$d_{k,j,m} = c_{k,j-1,m} \varepsilon^{i (\ell_{j,k-1,m,k} - \ell_{j,k,m,k})} \mathbf{p} - c_{k,j,m}$$

and

$$\ell_{k,j,m} = \ell_{j,k,m,k}.$$

**Definition 3.4 (Weierstrass Iterated Fractal Drum (Weierstrass IFD)).** We call Weierstrass Iterated Fractal Drum (in short, Weierstrass IFD), and denote by $\Gamma_W^\mathcal{I}$, the sequence of ordered pairs $(\Gamma_m, \varepsilon_m^m)_{m \in \mathbb{N}}$ where, for each $m \in \mathbb{N}$, $\Gamma_m$ is the $m^{th}$ prefractal approximation to the Weierstrass Curve $\Gamma_W$, as introduced in Definition 2.3, on page 11, and where $\varepsilon_m^m$ is the $m^{th}$ cohomology infinitesimal, as introduced in Definition 3.1, on page 30 above. Note that the $m^{th}$ prefractal graph approximation (viewed as an oriented graph) determines the $m^{th}$ prefractal curve (viewed as an oriented polygonal curve), and conversely. Indeed, the line segments of which the latter polygonal curve is comprised are nothing but the edges of the former prefractal graph.

In the sequel, $(\varepsilon^m)_{m \in \mathbb{N}}$ stands for the intrinsic cohomology infinitesimal, as introduced in Definition 3.1, on page 30 above.

**Definition 3.5 (Oscillatory period).** Following [42], [43], [39], we introduce the oscillatory period of the Weierstrass IFD:

$$\mathbf{p} = \frac{2\pi}{\ln N_b}.$$
Our results on fractal cohomology obtained in [DL22c] have highlighted the part played by specific threshold values for the number $\epsilon > 0$ at any step $m \in \mathbb{N}$ of the prefractal graph approximation; namely, the $m^{th}$ cohomology infinitesimal introduced in Definition 3.1, on page 37 just below.

**Definition 3.1 ($m^{th}$ Cohomology Infinitesimal).**

From now on, given any $m \in \mathbb{N}$, we will call $m^{th}$ cohomology infinitesimal the number $\epsilon_m^m > 0$ which also corresponds to the elementary horizontal length introduced in part i. in Definition 2.4, on page 15; i.e., $\epsilon_m^m = (\epsilon_m)^m = \frac{1}{N_b - 1} \frac{1}{N_b^m}$.

Observe that, clearly, $\epsilon_m$ itself – and not just $\epsilon_m^m$ – depends on $m$.

In addition, since $N_b > 1$, $\epsilon_m^m$ satisfies the following asymptotic behavior,

$$\epsilon_m^m \to 0, \text{ as } m \to \infty,$$

which, naturally, results in the fact that the larger $m$, the smaller $\epsilon_m^m$. It is for this reason that we call $\epsilon_m^m$ – or rather, the infinitesimal sequence $(\epsilon_m^m)_{m=0}^{\infty}$ of positive numbers tending to zero as $m \to \infty$, with $\epsilon_m^m = (\epsilon_m)^m$, for each $m \in \mathbb{N}$ – an infinitesimal. Note that this $m^{th}$ cohomology infinitesimal is the one naturally associated to the scaling relation of Property 2.6, on page 16.

In the sequel, it is also useful to keep in mind that the sequence of positive numbers $(\epsilon_m)_{m=0}^{\infty}$ itself satisfies

$$\epsilon_m \sim \frac{1}{N_b}, \text{ as } m \to \infty ;$$

i.e., $\epsilon_m \to \frac{1}{N_b}$, as $m \to \infty$. In particular, $\epsilon_m \not\to 0$, as $m \to \infty$, but, instead, $\epsilon_m$ tends to a strictly positive and finite limit.

**Remark 3.1 (Addressing Numerical Estimates).**

From a practical point of view, an important question is the value of the ratio

$$\frac{\text{Cohomology infinitesimal}}{\text{Maximal height}} = \frac{\epsilon_m^m}{h_m} ;$$

see relation (R6), on page 15.

Thanks to the estimates given in relation (R9), on page 27, we have that

$$\frac{\epsilon_m^m}{h_m} = L_m^{1-D_W} \mathcal{O}(1) = \epsilon_m^{m(1-D_W)} \mathcal{O}(1),$$

with

$$0 < C_{inf} \le \mathcal{O}(1) \le C_{sup}.$$
By definition, it coincides with the oscillatory period of the complexified Weierstrass function $W_{comp}$ obtained in Theorem 3.2 above, on page 33.

**Definition 3.6** ($\ell^{th}$-Order vibration mode). Given $\ell \in \mathbb{Z}$, we define the $\ell^{th}$ order vibration mode as the one associated to $\ell \mathbf{p}$.

**Definition 3.7** ($(m, \varepsilon^m)$-Upper and lower neighborhoods). Given $m \in \mathbb{N}$, and a point $M = (x, y) \in \mathbb{R}^2$, we denote by $d(M, \Gamma_m)$ the distance from $M$ to $\Gamma_m$.

Then, for all $t \ge 0$, we introduce the following subsets of $\mathbb{R}^2$:

i. The $(m, t)$-upper neighborhood of the $m^{th}$ prefractal approximation $\Gamma_m$:

$$ \mathcal{D}^{+}(\Gamma_m, t) = \left\{ M = (x, y) \in \mathbb{R}^2, x_{j,m} \le x \le x_{j+1,m}, 0 \le j \le \#V_m - 2, \text{ with } M \text{ located above } [M_{j,m}, M_{j+1,m}] \text{ and } d(M, \Gamma_m) \le t \right\}. $$

ii. The $(m, t)$-lower neighborhood of the $m^{th}$ prefractal approximation $\Gamma_m$:

$$ \mathcal{D}^{-}(\Gamma_m, t) = \left\{ M = (x, y) \in \mathbb{R}^2, x_{j,m} \le x \le x_{j+1,m}, 0 \le j \le \#V_m - 2, \text{ with } M \text{ located under } [M_{j,m}, M_{j+1,m}] \text{ and } d(M, \Gamma_m) \le t \right\}. $$

**Definition 3.8** ($(m, \varepsilon_m^m)$-Neighborhood). Given $m \in \mathbb{N}^*$ sufficiently large, along with $\mathcal{D}^{-}(\Gamma_m, \varepsilon_m^m)$ and $\mathcal{D}^{+}(\Gamma_m, \varepsilon_m^m)$ (as in Definition 3.7, on page 37 just above), we define the $(m, \varepsilon_m^m)$-Neighborhood as the union of the upper and lower ones, as follows:

$$ \mathcal{D}(\Gamma_m, \varepsilon_m^m) = \mathcal{D}^{-}(\Gamma_m, \varepsilon_m^m) \cup \mathcal{D}^{+}(\Gamma_m, \varepsilon_m^m). $$

Previous works provide a very unfriendly expression for the absolute value of the elementary heights, $|h_{j,m}|$, for $\frac{3N_b - 1}{4} \le j < N_b - 1$, and $(i_1, \dots, i_m) \in \{0, \dots, N_b - 1\}^m$, as

$$ |h_{j,m}| = |\lambda^m (y_{j+1} - y_j) - 2 \sum_{k=1}^{m} \lambda^{m-k} \sin \left( \frac{\pi}{N_b^{k+1}} \frac{1}{(N_b-1)} \right) \sin \left( \frac{\pi (2j+1)}{N_b^{k+1}} \frac{1}{(N_b-1)} + 2\pi \sum_{q=0}^{k} \frac{i_{m-q}}{N_b^{k-q}} \right) | $$

This is no longer the case, thanks to the series expansion of the Weierstrass function given in relation (16) in Theorem 3.2, on page 33.

Although it suffices to compute the Minkowski dimension of the Curve, one also requires, in the present work, an explicit expression for the elementary lengths $L_m, m \in \mathbb{N}^*$.
Given $q \in \mathbb{N}^*$, we then have

$$ \frac{1}{10^q} C_{inf} \le \frac{\varepsilon_m^m}{h_m} \le \frac{1}{10^q} C_{sup} $$

when

$$ \frac{C_{inf}}{10^q} \le e^{(1-D_W) \ln L_m} \le \frac{C_{sup}}{10^q}, $$

or, equivalently, when

$$ -\frac{1}{\ln N_b} \ln \left( (N_b - 1) \left( \frac{C_{sup}}{10^q} \right)^{\frac{1}{1-D_W}} \right) \le m \le -\frac{1}{\ln N_b} \ln \left( (N_b - 1) \left( \frac{C_{inf}}{10^q} \right)^{\frac{1}{1-D_W}} \right). $$

Numerical values for $N_b = 3$ and $\lambda = \frac{1}{2}$ yield:

i. For $q = 1$: $2 \le m \le 3$.

ii. For $q = 2$: $7 \le m \le 9$.

iii. For $q = 3$: $13 \le m \le 15$.

Hence, when $m$ increases, the ratio decreases, and tends to 0. This numerical – but very practical and explicit argument – also applies to our forthcoming neighborhoods, of width equal to the cohomology infinitesimal.

**Definition 3.2 (Iterated Fractal Drums (IFDs)).**

Let us consider a fractal curve $\mathcal{F} \subset \mathbb{R}^2$, obtained by means of a suitable IFS $\mathcal{T}_{\mathcal{F}}$ (consisting, in particular, of a family of $C^\infty$ maps from $\mathbb{R}^2$ to $\mathbb{R}^2$). For each $m \in \mathbb{N}$, we denote by $\mathcal{F}_m$ the $m^{th}$ prefractal approximation to the fractal $\mathcal{F}$. We restrict ourselves to the case when there exists a natural scaling relation associated to the sequence $(\Gamma_{\mathcal{F}_m})_{m \in \mathbb{N}}$, involving a sequence of elementary lengths (or cohomology infinitesimals) $(\varepsilon_{m,\mathcal{F}}^m)_{m \in \mathbb{N}}$, and, as in Definition 3.1, on page 37 above.

We then call *Iterated Fractal Drum* (in short, *IFD*), and denote by $\mathcal{F}^I$, the sequence of ordered pairs $(\mathcal{F}_m, \varepsilon_{m,\mathcal{F}}^m)_{m \in \mathbb{N}}$, where, for each $m \in \mathbb{N}$, $\mathcal{F}_m$ is the $m^{th}$ prefractal (graph) approximation associated with the fractal $\mathcal{F}$.

**Definition 3.3 (Weierstrass Iterated Fractal Drum (Weierstrass IFD)).**

We call *Weierstrass Iterated Fractal Drum* (in short, *Weierstrass IFD*), and denote by $\Gamma_{\mathcal{W}}^I$, the sequence of ordered pairs $(\Gamma_{\mathcal{W}_m}, \varepsilon_m^m)_{m \in \mathbb{N}}$ where, for each $m \in \mathbb{N}$, $\Gamma_{\mathcal{W}_m}$ is the $m^{th}$ prefractal approximation to the Weierstrass Curve $\Gamma_{\mathcal{W}}$, as introduced in Definition 2.2, on page 10, and where $\varepsilon_m^m$ is the $m^{th}$ cohomology infinitesimal, as introduced in Definition 3.1, on page 37 above. Note that the $m^{th}$ prefractal graph approximation (viewed as an oriented curve) determines the $m^{th}$ prefractal curve (viewed as an oriented polygonal curve), and conversely. Indeed, the line segments of which the latter polygonal curve is comprised are nothing but the edges of the former prefractal graph.

In the sequel, $(\varepsilon_m^m)_{m \in \mathbb{N}}$ stands for the cohomology infinitesimal, as introduced in Definition 3.1, on page 37 above.
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** The $(1, \epsilon_1^m)$-upper and lower neighborhoods, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$. The upper neighborhood is on the left, while the lower neighborhood is on the right. The prefractal graph $\Gamma_1$ is in red. For the sake of readability and understanding, different scales have been used for the horizontal and the vertical directions. (For colors in the figure(s), the reader is referred to the web version of this article.)</figcaption></figure>

The $(m, \epsilon_m^m)$-upper and lower neighborhoods introduced in Definition 3.7, on page 37, are then obtained by means of rectangles and wedges, as depicted in Figs. 10–14 (on pages 38–41).

**Proposition 3.4** ($(m, \epsilon_m^m)$-Upper neighborhood). According to Property 2.7, on page 15 (and Definition 2.5, on page 12), given a strictly positive integer $m$, the $(m, \epsilon_m^m)$-upper neighborhood consists of:

i. $(N_b - 1) N_b^m$ **rectangles**, each of length $\ell_{j-1,j,m}$, for $1 \le j \le N_b^m - 1$, and height $\epsilon^m$. Those rectangles are also **overlapping ones**, at least along their bottom. If we denote by $M_{j,m}$ the common vertex between two consecutive overlapping rectangles (see Fig. 13, on page 41), the area that is thus counted twice corresponds to parallelograms, of height $\epsilon_m^m$ and basis $\epsilon_m^m$ cotan $(\pi - \theta_{j-1,m} - \theta_{j,m})$; i.e., this area is equal to $(\epsilon_m^m)^2$ cotan $(\theta_{j-1,m} + \theta_{j,j+1,m})$.

Since one deals here with an upper neighborhood, one also has to subtract the areas of the **extra outer lower triangles**, i.e., $\frac{1}{2} \epsilon_m^m (b_{j-1,m} + b_{j,m})$, where, for $j$ in $\{1, \dots, (N_b - 1) N_b^m - 1\}$, $b_{j-1,m} \le h_{j-1,m}$ denotes the basis of the parallelogram in common to overlapping rectangles associated to the vertex $M_{j,m}$.

ii. $N_b^m r_b^+$ **upper wedges**, with $r_b^+ \in \mathbb{N}$, $1 \le r_b^+ \le N_b - 1$ (those wedges are to be understood in the strict sense, which means that the extreme ones are not taken into account here). If we denote by $M_{j,m}$ the vertex from which is issued the wedge (see Fig. 14, on page 41), the area of this latter wedge is given by
**Definition 3.4** ($m, \varepsilon_m^m$)-Upper and Lower Neighborhoods).

Given $m \in \mathbb{N}$, and a point $M \in \mathbb{R}^2$, we denote by $d(M, \Gamma_{W_m})$ the distance from $M$ to $\Gamma_{W_m}$. Then, for any sufficiently large $m$ (so that $\varepsilon_m^m$ be a sufficiently small positive number), we introduce:

i. The $(m, \varepsilon_m^m)$-Upper Neighborhood of the $m^{th}$ prefractal approximation $\Gamma_{W_m}$:

$$ \mathcal{D}^{+}(\Gamma_{W_m}, \varepsilon_m^m) = \{M = (x, y) \in \mathbb{R}^2, y \geq W(x) \text{ and } d(M, \Gamma_{W_m}) \leq \varepsilon_m^m\}; $$

ii. The $(m, \varepsilon_m^m)$-Lower Neighborhood of the $m^{th}$ prefractal approximation $\Gamma_{W_m}$:

$$ \mathcal{D}^{-}(\Gamma_{W_m}, \varepsilon_m^m) = \{M = (x, y) \in \mathbb{R}^2, y \leq W(x) \text{ and } d(M, \Gamma_{W_m}) \leq \varepsilon_m^m\}. $$

**Definition 3.5** ($m, \varepsilon_m^m$)-Neighborhood).

Given $m \in \mathbb{N}$ sufficiently large (as in Definition 3.4, on page 39 just above), along with $\mathcal{D}^{-}(\Gamma_{W_m}, \varepsilon_m^m)$ and $\mathcal{D}^{+}(\Gamma_{W_m}, \varepsilon_m^m)$, we define the $(m, \varepsilon_m^m)$-Neighborhood as the union of the upper and lower ones, as follows:

$$ \mathcal{D}(\Gamma_{W_m}, \varepsilon_m^m) = \mathcal{D}^{-}(\Gamma_{W_m}, \varepsilon_m^m) \cup \mathcal{D}^{+}(\Gamma_{W_m}, \varepsilon_m^m). $$

**Definition 3.6** (Left-Side and Right-Side $(m, \varepsilon_m^m)$-Neighborhoods).

Given $m \in \mathbb{N}$ sufficiently large, we introduce:

i. the Left-Side $(m, \varepsilon_m^m)$-Neighborhood of the $m^{th}$ prefractal approximation $\Gamma_{W_m}$ as

$$ \mathcal{D}_{\text{Left}}(\Gamma_{W_m}, \varepsilon_m^m) = \{M = (x, y) \in [0, \frac{1}{2}] \times \mathbb{R}, d(M, \Gamma_{W_m}) \leq \varepsilon_m^m\}; $$

ii. the Right-Side $(m, \varepsilon_m^m)$-Neighborhood of the $m^{th}$ prefractal approximation $\Gamma_{W_m}$ as

$$ \mathcal{D}_{\text{Right}}(\Gamma_{W_m}, \varepsilon_m^m) = \{M = (x, y) \in [\frac{1}{2}, 1] \times \mathbb{R}, d(M, \Gamma_{W_m}) \leq \varepsilon_m^m\}. $$

Those neighborhoods are symmetric with respect to the vertical line $x = \frac{1}{2}$; see Figure 5, on page 17, and Figure 13, on page 46. They constitute, in a sense, a partition of the whole tubular neighborhood.

**Notation 9** (Integer to Cohomology Infinitesimal Map).

We hereafter introduce the map

$$ \varepsilon_m^m \mapsto m(\varepsilon_m^m) = [-\ln_{N_b}((N_b - 1)\varepsilon_m^m)] = [m(\varepsilon_m^m)] + \{m(\varepsilon_m^m)\}, $$
<figure><img src="image_14.png" /><figcaption>Fig. 11. The $(2, \varepsilon_2^2)$-neighborhood, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$. For the sake of readability and understanding, different scales have been used for the horizontal and the vertical directions.</figcaption></figure>

$$ \frac{1}{2} (\pi - \theta_{j-1,m} - \theta_{j,m}) (\varepsilon_m^m)^2, \quad \text{for } 1 \le j \le N_b^m - 1. $$

The number of upper wedges $N_b^m r_b^+$ is determined by the shape of the initial polygon $P_0$, as well by the existence of reentrant angles. This directly follows from Property 2.19, on page 27. It is given by

$$ N_b^m r_b^+ = N_b^m N_b - 2 N_b^m \left[ \frac{N_b - 3}{4} \right] - 1. \quad (24) $$

iii. Two **extreme wedges** (see Fig. 15), each of area equal to $\frac{1}{2} \pi (\varepsilon_m^m)^2$.

**Proposition 3.5** ($(m, \varepsilon^m)$-Lower neighborhood). In the same way, given a strictly positive integer $m$, the $(m, \varepsilon_m^m)$-lower neighborhood consists of:

i. $(N_b - 1) N_b^m$ **rectangles**, each of length $\ell_{j-1,j,m}$, for $1 \le j \le N_b^m - 1$, and height $\varepsilon_m^m$. Those rectangles are also **overlapping ones**, this time at least along their top. If we denote by $M_{j,m}$ the common vertex between two consecutive overlapping rectangles, the area that is thus counted twice again corresponds to parallelograms,
where [·] and {·} respectively denote the integer and fractional parts.

For notational simplicity, we temporarily set $x = m(\varepsilon_m^m) = -\ln_{N_b}((N_b - 1)\varepsilon_m^m)$.

Previous works give a very unfriendly expression for the absolute value of the elementary heights, $|h_{j,m}|$, for $\frac{3N_b - 1}{4} \le j < N_b - 1$, and $(i_1, \dots, i_m) \in \{0, \dots, N_b - 1\}^m$, as

$$|h_{j,m}| = \left| \lambda^m (y_{j+1} - y_j) - 2 \sum_{k=1}^{m} \lambda^{m-k} \sin\left(\frac{\pi}{N_b^{k+1}(N_b-1)}\right) \sin\left(\frac{\pi(2j+1)}{N_b^{k+1}(N_b-1)} + 2\pi \sum_{q=0}^{k} \frac{i_{m-q}}{N_b^{k-q}}\right) \right|.$$

Although it is sufficient to compute the Minkowski dimension of the Curve, one also requires, in the present work, an explicit expression for the elementary lengths $L_m, m \in \mathbb{N}^*$.

Accordingly, one has

$$N_b^m = N_b^{[x]} = N_b^{x-\{x\}} \quad , \quad N_b^{-m} = N_b^{\{x\}-x},$$

$$L_m = \frac{1}{(N_b - 1) N_b^m} = \frac{1}{N_b - 1} N_b^{\{x\}-x},$$

$$-([x] + \{x\}) \ln N_b = \ln((N_b - 1)\varepsilon_m^m) \quad , \quad e^{-([x]+\{x\}) \ln N_b} = (N_b - 1)\varepsilon_m^m \quad , \quad \varepsilon_m^m = \frac{1}{N_b - 1} N_b^{-([x]+\{x\})}.$$

The $(m, \varepsilon_m^m)$-Upper and Lower Neighborhoods introduced in Definition 3.4, on page 39, are then obtained by means of rectangles and wedges, as depicted in Figures 8-14 (on pages 42-47).

**Proposition 3.1** ($(m, \varepsilon_m^m)$-Upper Neighborhood).

According to Property 2.4, on page 14 (and Definition 2.4, on page 15), given a strictly positive integer $m$, the $(m, \varepsilon_m^m)$-upper neighborhood consists of:

i. $(N_b - 1) N_b^m$ **rectangles**, each of length $\ell_{j-1,j,m}$, for $1 \le j \le N_b^m - 1$, and height $\varepsilon_m^m$.

Those rectangles are also **overlapping ones**, at least at their bottom. If we denote by $M_{j,m}$ the common vertex between two consecutive overlapping rectangles (see Figure 10, on page 44), the area that is thus counted twice corresponds to parallelograms, of height $\varepsilon_m^m$ and basis $\varepsilon_m^m \cotan(\pi - \theta_{j-1,j,m} - \theta_{j,j+1,m})$; i.e., this area is equal to $(\varepsilon_m^m)^2 \cotan(\theta_{j-1,j,m} + \theta_{j,j+1,m})$.

Since one deals here with an upper neighborhood, one also has to subtract the areas of the **extra outer lower triangles**, i.e., $\frac{1}{2} \varepsilon_m^m (b_{j-1,j,m} + b_{j,j+1,m})$.

ii. $N_b^m \left(1 + 2\left[\frac{N_b - 3}{4}\right]\right) - 1$ **upper wedges** (to be understood in the strict sense, which means that the extreme ones are not taken into account here). If we denote by $M_{j,m}$ the vertex from which is issued the wedge (see Figure 14, on page 47), the area of this latter wedge is given by
<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Consecutive vertices of the $(m, \varepsilon_m^m)$-upper neighborhood, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$. The prefractal graph $\Gamma_m$ is in red. For the sake of readability and understanding, different scales have been used for the horizontal and the vertical directions.</figcaption></figure>

of height $\varepsilon_m^m$ and basis $\varepsilon_m^m$ cotan $(\pi - \theta_{j-1,m} - \theta_{j,m})$; i.e., this area is equal to $(\varepsilon_m^m)^2$ cotan $(\theta_{j-1,m} + \theta_{j,j+1,m})$.

Since one deals here with a lower neighborhood, one has this time to subtract the areas of the **extra outer upper triangles**, namely, amounting to $\frac{1}{2}\varepsilon_m^m(b_{j-1,m}+b_{j,j+1,m})$, where, for $j$ in $\{1, \dots, (N_b-1)N_b^m-1\}$, $b_{j-1,m}$ denotes the basis of the parallelogram in common to overlapping rectangles associated to the vertex $M_{j,m}$.

ii. $N_b^m r_b^-$ **lower wedges**, with $r_b^- \in \mathbb{N}$, $1 \le r_b^- \le N_b - 1$. If we denote by $M_{j,m}$ the vertex from which is issued the wedge, the area of this latter wedge is obtained as previously, and is given by

$$ \frac{1}{2} (\pi - \theta_{j-1,m} - \theta_{j,m}) (\varepsilon_m^m)^2, \quad \text{for } 1 \le j \le N_b^m - 2. $$
$$ \frac{1}{2} (\pi - \theta_{j-1,j,m} - \theta_{j,j+1,m}) (\varepsilon_m^m)^2, \quad \text{for } 1 \le j \le N_b^m - 2. $$

The number of wedges is determined by the shape of the initial polygon $P_0$, as well as by the existence of reentrant angles. This directly follows from Property 2.18, on page 29. For the sake of simplicity, we set

$$ r_b^+ = 1 + 2 \left[ \frac{N_b - 3}{4} \right]. \qquad (\mathcal{R} 14) $$

iii. Two **extreme wedges** (see Figure 15, on page 48), each of area equal to $\frac{1}{2} \pi (\varepsilon_m^m)^2$.

**Proposition 3.2** ($m, \varepsilon_m^m$-Lower Neighborhood).

In the same way, given a strictly positive integer $m$, the $(m, \varepsilon_m^m)$-lower neighborhood consists of:

i. $(N_b - 1) N_b^m$ **rectangles**, each of length $\ell_{j-1,j,m}$, for $1 \le j \le N_b^m - 1$, and height $\varepsilon_m^m$.

Those rectangles are also **overlapping ones**, this time at least at their top. If we denote by $M_{j,m}$ the common vertex between two consecutive overlapping rectangles, the area that is thus counted twice again corresponds to parallelograms, of height $\varepsilon_m^m$ and basis $\varepsilon_m^m$ cotan $(\pi - \theta_{j-1,j,m} - \theta_{j,j+1,m})$; i.e., this area is equal to $(\varepsilon_m^m)^2$ cotan $(\theta_{j-1,j,m} + \theta_{j,j+1,m})$.

Since one deals here with a lower neighborhood, one has this time to subtract the areas of the **extra outer upper triangles**, namely, amounting to $\frac{1}{2} \varepsilon_m^m (b_{j-1,j,m} + b_{j,j+1,m})$.

ii. $N_b^m (N_b - 2 \left[ \frac{N_b - 3}{4} \right]) - 1$ **lower wedges**. If we denote by $M_{j,m}$ the vertex from which is issued the wedge, the area of this latter wedge is obtained as previously, and is given by

$$ \frac{1}{2} (\pi - \theta_{j-1,j,m} - \theta_{j,j+1,m}) (\varepsilon_m^m)^2, \quad \text{for } 1 \le j \le N_b^m - 2. $$

The number of lower wedges is determined by the shape of the initial polygon $P_0$, as well as by the existence of reentrant angles. This directly comes from Property 2.18, on page 29. For the sake of simplicity, we set

$$ r_b^- = N_b - 2 \left[ \frac{N_b - 3}{4} \right]. \qquad (\mathcal{R} 15) $$
<figure><img src="image_16.png" /><figcaption>Fig. 13. Two overlapping rectangles, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$. For the sake of readability and understanding, different scales have been used for the horizontal and the vertical directions.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 14. An upper wedge.</figcaption></figure>

Similarly, the number of lower wedges $N_b^m r_b^-$ is determined by the shape of the initial polygon $P_0$, as well as by the existence of reentrant angles. This directly comes from Property 2.19, on page 27. It is given by

$$N_b^m r_b^- = N_b^m N_b - 2 N_b^m \left[ \frac{N_b - 3}{4} \right]. \qquad (25)$$
Remark 3.2.

i. The number of upper overlapping rectangles is equal to the number of lower extra triangles, and also to the number of upper wedges.

ii. The number of lower overlapping rectangles is equal to the number of upper extra triangles, and also to the number of lower wedges.

iii. In light of i. and ii. just above, those numbers can be respectively calculated as being equal to

$$ (r_b^+ - 1) N_b^m \quad \text{and} \quad (r_b^- - 1) N_b^m, $$

where the coefficients $r_b^-$ and $r_b^+$ are respectively defined in formulas (R15), page 41 and (R14), page 41.

iv. Note that the small parameter $\epsilon_m^m$ has to be sufficiently small (say $0 < \epsilon_m^m < \epsilon_{m_0}^{m_0}$, for some $\epsilon_{m_0}^{m_0} > 0$ which exists, but appears difficult to specify explicitly) in order to avoid more unfriendly overlaps than the parallelograms; see Figure 16, on page 49.

<figure><img src="image_9.png" /><figcaption>Figure 8: The $(1, \epsilon_1^1)$-Upper and Lower Neighborhoods, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>**Fig. 15.** The extreme wedges, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$. For the sake of readability and understanding, different scales have been used for the horizontal and the vertical directions.</figcaption></figure>

**Remark 3.2.**

i. The number of upper overlapping rectangles is equal to the number of lower extra triangles, and also to the number of upper wedges.

ii. The number of lower overlapping rectangles is equal to the number of upper extra triangles, and also to the number of lower wedges.

iii. In light of i. and ii. just above, those numbers can be respectively calculated as being equal to
<figure><img src="image_10.png" /><figcaption>Figure 9: The $(1, \varepsilon_1^1)$-Upper Neighborhood, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$.</figcaption></figure>
$$ (r_b^+ - 1) N_b^m \quad \text{and} \quad r_b^- N_b^m, $$

where the coefficients $r_b^-$ and $r_b^+$ are respectively given in formulas (25), page 41 and (24), page 39.

iv. Note that the parameter $\epsilon_m^m$ has to be sufficiently small [say $0 < \epsilon_m^m < \epsilon_{m_0}^{m_0}$, for some $m_0 \in \mathbb{N}^*$ sufficiently large and optimal (i.e., smallest) which exists, but appears difficult to specify explicitly] in order to avoid more unfriendly overlaps than those occurring for the parallelograms.

**Proposition 3.6 (Contribution of the rectangles to the tubular volume).** Given $m \in \mathbb{N}^*$ sufficiently large, the (exact) contribution of the $(N_b - 1) N_b^m$ rectangles to the tubular volume is given by

$$ \begin{aligned} V_{m, \text{Rectangles}} &= 2 \sum_{j=1}^{\#V_m - 1} \epsilon_m^m \ell_{j-1,m} \\ &= 2 \mathcal{R}e \left( \sum_{j=1}^{\#V_m - 1} \epsilon_m^m C_{\ell, j-1, m} \sum_{k=0}^{m} d_{k, j, m} \epsilon^{k(2-D_W)} \epsilon^{i \ell_{k, j, m, k} \mathbf{P}} \right) \\ &= 2 \mathcal{R}e \left( \sum_{j=1}^{\#V_m - 1} \frac{\epsilon_m^m}{N_b - 1} C_{\ell, j-1, m} \sum_{k=0}^{m} d_{k, j, m} \epsilon^{k(2-D_W)} \epsilon^{i \ell_{k, j, m} \mathbf{P}} \right) \quad (26) \\ &= \sum_{j=1}^{\#V_m - 1} \frac{\epsilon_m^m}{N_b - 1} C_{\ell, j-1, m} \\ &\quad \times \sum_{k=0}^{m} \epsilon^{k(2-D_W)} \left( \overline{d_{k, j, m}} \epsilon^{-i \ell_{k, j, m} \mathbf{P}} + d_{k, j, m} \epsilon^{i \ell_{k, j, m} \mathbf{P}} \right), \end{aligned} $$

where the (strictly positive) constants $C_{\ell, j-1, m}$ are defined in relation (10) of part ii. of Corollary 2.16, on page 25, while the coefficients $d_{k, j, m}$ and the positive integers $\ell_{k, j, m}$ are given in Corollary 3.3, on page 36.

Note that the contribution of the rectangles to the tubular volume is, geometrically, the main one. For this reason, we have used the capital letter R, contrary to the other – and forthcoming – contributions.

**Proof.** We simply apply the estimates obtained in relation (10), given on page 25. □

**Proposition 3.7 (Contribution of the extreme, upper and lower wedges to the tubular volume).**

i. Given $m \in \mathbb{N}^*$ sufficiently large, the (exact) contribution of the **extreme wedges** to the tubular volume is given by
<figure><img src="image_11.png" /><figcaption>Figure 10: Two overlapping rectangles, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$.</figcaption></figure>
$$ \mathcal{V}_{m, \text{extreme wedges}} = \pi (\varepsilon_m^m)^2 = \frac{\pi \varepsilon^{2m}}{(N_b - 1)^2}. $$

ii. Given $m \in \mathbb{N}^*$ sufficiently large, the (exact) contribution of the $r_b^+ N_b^m$ **upper wedges** to the tubular volume is given by

$$ \begin{aligned} \mathcal{V}_{m, \text{upper wedges}} &= \frac{1}{2} \sum_{j=1}^{\#V_m - 1} (\pi - \theta_{j-1,m} - \theta_{j,m}) (\varepsilon_m^m)^2 \\ &= \frac{1}{2} \sum_{j=1}^{\#V_m - 1} (\varepsilon_m^m)^2 \left( \pi - \arctan \frac{L_m}{h_{j-1,m}} - \arctan \frac{L_m}{h_{j,m}} \right) \\ &= \frac{(\varepsilon_m^m)^2}{2} r_b^+ N_b^m \left( \pi - (C_{h,j-1,m}^{-1} + C_{h,j,m}^{-1}) \varepsilon_m^{m(D_W-1)} \right) \\ &= \frac{\varepsilon^{2m}}{(N_b - 1)^2} r_b^+ N_b^m \left( \pi - (C_{h,j-1,m}^{-1} + C_{h,j,m}^{-1}) \frac{\varepsilon^{m(D_W-1)}}{(N_b - 1)^{D_W-1}} \right), \end{aligned} \quad (27) $$

where the coefficients $C_{h,j-1,m}$ and $C_{h,j,m}$ are given in relation (8) of part i. of Corollary 2.16, given on page 25.

iii. In the same way, given $m \in \mathbb{N}^*$ sufficiently large, the (exact) contribution of the $r_b^- N_b^m - 1$ **lower wedges** to the tubular volume is given by

$$ \begin{aligned} \mathcal{V}_{m, \text{lower wedges}} &= \frac{(\varepsilon_m^m)^2}{2} r_b^- N_b^m \left( \pi - (C_{h,j-1,m}^{-1} + C_{h,j,m}^{-1}) \varepsilon_m^{m(D_W-1)} \right) \\ &= \frac{\varepsilon^{2m}}{(N_b - 1)^2} r_b^- N_b^m \left( \pi - (C_{h,j-1,m}^{-1} + C_{h,j,m}^{-1}) \frac{\varepsilon^{m(D_W-1)}}{(N_b - 1)^{D_W-1}} \right), \end{aligned} \quad (28) $$

where, as above, the coefficients $C_{h,j-1,m}$ and $C_{h,j,m}$ are given in relation (8), given on page 25.

**Proposition 3.8** (Negative contribution of the extra outer triangles to the tubular volume). Given $m \in \mathbb{N}^*$ sufficiently large, the negative contribution of the **extra outer lower and upper triangles** to the tubular volume is given by

$$ \begin{aligned} \mathcal{V}_{m, \text{extra outer triangles}} &= -\frac{\varepsilon_m^m}{2} \sum_{j=1}^{\#V_m - 1} \{b_{j-1,m} + b_{j,j+1,m}\} \\ &= -\operatorname{Re} \left( \sum_{j=1}^{\#V_m - 1} \frac{\varepsilon^m}{2(N_b - 1)} (C_{b,j-1,m} + C_{b,j,m}) \sum_{k=0}^{m} d_{k,j,m} \varepsilon^{k(2-D_W)} \varepsilon^{i \ell_{k,j,m} \mathbf{P}} \right) \end{aligned} \quad (29) $$
<figure><img src="image_12.png" /><figcaption>Figure 11: The $(1, \varepsilon_1^1)$, $(2, \varepsilon_2^2 m)$ and $(3, \varepsilon_m^m)$-Neighborhoods, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 12: The $(1, \varepsilon_1^1)$, $(2, \varepsilon_2^2)$ and $(3, \varepsilon_3^3)$-Upper Neighborhoods, in the case when $\lambda = \frac{1}{2}$ and $N_b = 4$.</figcaption></figure>
$$ = - \sum_{j=1}^{\#V_m-1} \frac{\varepsilon^m}{4(N_b-1)} (C_{b,j-1,m} + C_{b,j,m}) \\ \times \sum_{k=0}^{m} \varepsilon^{k(2-D_W)} (\overline{d_{k,j,m}} \varepsilon^{-i \ell_{k,j,m} \mathbf{P}} + d_{k,j,m} \varepsilon^{i \ell_{k,j,m} \mathbf{P}}), $$

where the (strictly positive) constants $C_{b,j-1,m}$ are defined in Lemma 3.9 below, on page 45, while the coefficients $d_{k,j,m}$ and the positive integers $\ell_{k,j,m}$ are given in Corollary 3.3, on page 36.

**Lemma 3.9 (Basis of the parallelograms in common to overlapping rectangles).** Given $m \in \mathbb{N}^*$, and $j$ in $\{0, \dots, (N_b - 1) N_b^m - 2\}$, the basis $b_{j,m}$ of the parallelogram in common to overlapping rectangles associated to the vertex $M_{j,m}$ is such that

$$ b_{j,m} = C_{b,j,m} L_m^{2-D_W}, $$

where $C_{b,j-1,m}$ depends on $j$ and $m$ and is such that

$$ 0 < C_{b,j-1,m} \le C_{sup}. $$

**Proof.** We simply use the fact that

$$ b_{j,m} \le h_{j,m}, $$

in conjunction with part i. of Corollary 2.16, on page 25. $\square$

**Proposition 3.10 (Negative contribution of the overlapping rectangles to the tubular volume).** Given $m \in \mathbb{N}^*$ sufficiently large, the negative contribution of the upper and lower overlapping rectangles to the tubular volume is given by

V_{m,upper and lower parallelograms}

$$ \begin{aligned} &= -\varepsilon_m^m \sum_{j=1}^{\#V_m-1} b_{j-1,m} \\ &= -\operatorname{Re} \left( \sum_{j=1}^{\#V_m-1} \varepsilon_m^m C_{b,j-1,m} \sum_{k=0}^{m} d_{k,j,m} \varepsilon^{k(2-D_W)} \varepsilon^{i \ell_{k,j,m} \mathbf{P}} \right) \tag{30} \\ &= -\sum_{j=1}^{\#V_m-1} \frac{C_{b,j-1,m} \varepsilon^m}{2(N_b-1)} \sum_{k=0}^{m} \varepsilon^{k(2-D_W)} \left( \overline{d_{k,j,m}} \varepsilon^{-i \ell_{k,j,m} \mathbf{P}} + d_{k,j,m} \varepsilon^{i \ell_{k,j,m} \mathbf{P}} \right), \end{aligned} $$

where the coefficients $d_{k,j,m}$ and the positive integers $\ell_{k,j,m}$ are given in Corollary 3.3, on page 36, and where the (strictly positive) constants $C_{b,j-1,m}$ have been introduced in Lemma 3.9, on page 45.
<figure><img src="image_14.png" /><figcaption>Figure 13: The $(3, \varepsilon_3^3)$-Left and Right-Side Neighborhoods, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$.</figcaption></figure>
**Remark 3.3** (*Connection between fractality and the cohomology infinitesimal*). As is mentioned in [12], the cohomology infinitesimal (or, equivalently, the elementary length) – which obviously depends on the magnification scale (i.e., the chosen prefractal approximation) – can be seen as a *transition scale* between the fractal domain and the classical (or Euclidean) one. In fact, we could say that the system is fractal below this scale, and classical above (for the level of magnification considered). In the limit when the integer *m* associated with the prefractal approximation tends to infinity, the system is fractal below the cohomological infinitesimal (which is really an infinitesimal, in this case), i.e., at small scales, and is classical beyond, i.e., on a large scale. Accordingly, in physicists’ language, we observe a natural *phase transition* between *Euclidean* and *fractal* prefractal approximations. Note that this is in perfect agreement with what is suggested by the French physicist Laurent Nottale in his work [48] about scale-relativity.

The Complex Dimensions of a fractal set characterize their intrinsic vibrational properties. Thus far, the values of the Complex Dimensions were obtained by studying the oscillations of a small neighborhood of the boundary, i.e., of a tubular neighborhood, where points are located within an epsilon distance from any edge. In the case of our fractal Weierstrass Curve $\Gamma_W$, which is, also, the limit of the sequence of (polygonal) prefractal approximations $(\Gamma_m)_{m \in \mathbb{N}}$, it is natural to envision the infinitesimal tubular neighborhood of $\Gamma_W$ associated with the cohomology infinitesimal $(\varepsilon_m^m)_{m \in \mathbb{N}}$, as the limit of the (obviously convergent) sequence $(\mathcal{D}(\Gamma_m, \varepsilon_m^m))_{m \in \mathbb{N}}$ of $\varepsilon_m^m$-neighborhoods of $\Gamma_m$, where, for each integer $m \in \mathbb{N}$, $\varepsilon_m^m$ is the $m^{th}$ cohomology infinitesimal introduced in Definition 3.1, on page 30 above.

**Property 3.11** (*The nested neighborhoods [14]*). i. Given $m \in \mathbb{N}$ sufficiently large, there exists $k_1 \in \mathbb{N}$ such that, for all $k \ge k_1$, the polygonal neighborhood $\mathcal{D}(\Gamma_m)$ introduced in Definition 2.6, on page 17 contains, but for a finite number of wedges and a finite number of parts of rectangles, the $(m+k, \varepsilon_{m+k}^m)$-neighborhood; see Fig. 5, on pages 18, along with Fig. 16, on page 47.

ii. Given $m \in \mathbb{N}$ sufficiently large, there exists $k_2 \in \mathbb{N}$ such that, for all $k \ge k_2$, the $(m, \varepsilon_m^m)$-neighborhood contains the polygonal neighborhood $\mathcal{D}(\Gamma_{m+k})$ introduced in Definition 2.6, on page 17.

iii. Given $m \in \mathbb{N}$ sufficiently large, there exists $k_3 \in \mathbb{N}$ such that, for all $k \ge k_3$, the polygonal neighborhood $\mathcal{D}(\Gamma_m)$ introduced in Definition 2.6, on page 17 contains the polygonal neighborhood $\mathcal{D}(\Gamma_{m+k})$.

For the detailed and technical proof of Property 3.11, we refer to our recent work [14].

**Corollary 3.12** (of Property 3.11, given on page 46 [14]). We immediately deduce from Property 3.11 that there exists $m_0 \in \mathbb{N}$ such that, for all $k \ge m_0$, the polyhedral neighborhood $\mathcal{D}(\Gamma_m)$ introduced in Definition 2.6, on page 17 contains the vertices of the prefractal approximation $\Gamma_{m+k}$ to the Weierstrass Curve; i.e., the set of vertices $V_{m+k}$.
<figure><img src="image_15.png" /><figcaption>Figure 14: An upper wedge.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Fig. 16. The exterior boundary of the polygonal neighborhood $\mathcal{D}(\Gamma_3)$ (in red in the online version, darker in the printed version), and the tubular neighborhood $\mathcal{D}(\Gamma_7, \varepsilon_7^m)$ (in cyan in the online version, lighter in the printed version), in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$.</figcaption></figure>

In particular, the density Property 2.5, on page 11, also ensures that the Weierstrass Curve is contained in $\mathcal{D}(\Gamma_m)$:

$$ \forall m \ge m_0 : \quad \Gamma_W \subset \mathcal{D}(\Gamma_m). $$

This ensures the existence of another integer $m_1 \in \mathbb{N}$ such that, for all $m \ge m_1$, the tubular neighborhood $\mathcal{D}(\Gamma_m, \varepsilon_m^m)$ contains the Weierstrass Curve $\Gamma_W$, thus being a neighborhood of $\Gamma_W$,

$$ \forall m \ge m_1 : \quad \Gamma_W \subset \mathcal{D}(\Gamma_{m+k}, \varepsilon_m^m). $$

**Proof.** The first part of the corollary is proved in [14].

The second part, i.e., the existence of an integer $m_1 \in \mathbb{N}$ such that, for all $m \ge m_1$, the tubular neighborhood $\mathcal{D}(\Gamma_m, \varepsilon_m^m)$ contains the Weierstrass Curve $\Gamma_W$, directly comes from the result given in part ii. of Property 3.11, given on page 46, which ensures the existence of an integer $k_1 \in \mathbb{N}$ such that, for all $k \ge k_1$, the $(m, \varepsilon_m^m)$-neighborhood contains the polygonal neighborhood $\mathcal{D}(\Gamma_{m+k})$:

$$ \forall m \ge m_1 : \quad \Gamma_W \subset \mathcal{D}(\Gamma_m) \subset \mathcal{D}(\Gamma_{m+k}, \varepsilon_{m+k}^{m+k}). $$

The desired result is obtained by letting $m_1 = m_0 + k_1$. $\square$
<figure><img src="image_16.png" /><figcaption>Figure 15: The extreme wedges, in the case when $\lambda = \frac{1}{2}$ and $N_b = 3$.</figcaption></figure>
## 4. Complex dimensions of the Weierstrass IFD

### 4.1. Prefractal tube formulas and prefractal effective zeta functions

In order to obtain the main results of this section – namely, Theorem 4.1, on page 48, and Theorem 4.3, on page 54, we consider the contribution to the (pre)fractal tube formulas brought by the various types of geometric elements in the $\varepsilon_m^m$-neighborhood of $\Gamma_m$. We stress the fact that, due to the above computations, the value of the $m^{th}$ cohomology infinitesimal $\varepsilon_m^m$ has to be sufficiently small. This means, in particular, that $m \in \mathbb{N}^*$ has to be sufficiently large, throughout this section.

**Theorem 4.1** (Prefractal tube formulas for the Weierstrass IFD). Given $m \in \mathbb{N}$ sufficiently large (see part iv. of Remark 3.2, on page 42), the $m^{th}$ total (volume) contribution to the tubular volume $V_m$ (or two-dimensional Lebesgue measure) of the $\varepsilon_m^m$-neighborhood of the $m^{th}$ prefractal approximation $\Gamma_m$, where $\varepsilon = (\varepsilon_m^m)_{m \in \mathbb{N}}$ is the cohomology infinitesimal, as introduced in Definition 3.1, on page 30, is given by the following prefractal tube formula:

$$ V_m = \varepsilon^m \sum_{j=1}^{\#V_m-1} C_{j-1,m} \sum_{k=0}^{m} \varepsilon^{k(2-D_W)} (\overline{d_{k,j,m}} \varepsilon^{-i \ell_{k,j,m} \mathbf{P}} + d_{k,j,m} \varepsilon^{-i \ell_{k,j,m} \mathbf{P}}) + \frac{\pi \varepsilon^{2m}}{2(N_b-1)^2}, \quad (31) $$

with $C_{j-1,m} > 0$, and where the coefficients $d_{k,j,m} \in \mathbb{C}$ and the integers $\ell_{k,j,m} \in \mathbb{N}$ are given in Corollary 3.3, on page 36.

For the sake of concision, we will hereafter write $V_m$ in the following form:

$$ V_m = \varepsilon^m \sum_{j=1}^{\#V_m-1} \sum_{k=0}^{m} \varepsilon^{k(2-D_W)} (\overline{C_{k,j,m}} \varepsilon^{-i \ell_{k,j,m} \mathbf{P}} + C_{k,j,m} \varepsilon^{-i \ell_{k,j,m} \mathbf{P}}) + \frac{\pi \varepsilon^{2m}}{2(N_b-1)^2}, \quad (32) $$

where, for $1 \le j \le \#V_m - 1$, $1 \le k \le m$, the nonzero complex coefficients $C_{k,j,m}$ are given by

$$ C_{k,j,m} = C_{j-1,m} d_{k,j,m} \in \mathbb{C}^*. \quad (33) $$

**Proof.** All of the main statements in the theorem concerning the $m^{th}$ tubular volume $V_m$ follow by combining Propositions 3.6–3.10, on pages 43–45 above:

$$ V_m = V_{m,\text{Rectangles}} + V_{m,\text{lower wedges}} + V_{m,\text{upper wedges}} + V_{m,\text{extreme wedges}} \\ + V_{m,\text{extra outer triangles}} + V_{m,\text{upper and lower parallelograms}}. $$
<figure><img src="image_17.png" /><figcaption>Figure 16: Two overlapping rectangles, when the parameter $\varepsilon_m^m$ is not sufficiently small: the overlap is a pentagon.</figcaption></figure>

In order to obtain, for the $\varepsilon$-tubular volume at a given level $m \in \mathbb{N}^*$ (i.e., the area of the $\varepsilon$-neighborhood of $\Gamma_{\mathcal{W}_m}$, the $m^{th}$ prefractal approximation) – not only a pleasant and understandable expression – but, also, an expression which takes into account the underlying fractality, we have to consider the case when $\varepsilon = \varepsilon_m^m$; i.e., the case when the infinitesimal is equal to the cohomology infinitesimal, as introduced in Definition 3.1, on page 37. Indeed, only the cohomology infinitesimal conveys the scaling relation associated with the Weierstrass IFD. This enables us to make use of the following relations, coming from Corollary 2.15, on page 27. Note that the associated tubular volume is, therefore, not a plain tubular volume associated with a plain polygonal graph. Since the cohomology infinitesimal is associated to the scaling relation obeyed by the Weierstrass function (see Property 2.6, on page 16), this tubular volume takes into account the fractality of the limit object of the sequence of prefractal graphs $(\Gamma_{\mathcal{W}_m})_{m \in \mathbb{N}}$, i.e., the Weierstrass Curve $\Gamma_W$.