Note that the area of the first extreme wedge – which is also the first upper wedge of the upper neighborhood – has to be subtracted from the total volume coming from the extreme wedges, the upper wedges, and the lower wedges. Indeed, otherwise, it is counted twice.

Thus, by construction, the identity (31) above, on page 48, holds. We also have that

$$C_{j-1,m} = C_{\ell,j-1,m} - \frac{3(C_{b,j-1,m} + C_{b,j,m})}{(N_b - 1)} > 0,$$

since the area of the tubular volume is strictly positive.

Then, the terms

$$\frac{\varepsilon^{2m}}{(N_b - 1)^2} (r_b^- + r_b^+) N_b^m \left( \pi - \left( C_{h,j-1,m}^{-1} + C_{h,j,m}^{-1} \right) \frac{t^{D_W}}{(N_b - 1)^{D_W}} \right),$$

which can be equivalently written as

$$\varepsilon^m \times \frac{\varepsilon^m}{(N_b - 1)^2} (r_b^- + r_b^+) N_b^m \left( \pi - \left( C_{h,j-1,m}^{-1} + C_{h,j,m}^{-1} \right) \frac{t^{D_W}}{(N_b - 1)^{D_W}} \right),$$

do not need to be taken into account in the computation of the associated (local) effective tube zeta function, since, when $m \to \infty$, they would lead to terms tending to 0 (because of the factor $\varepsilon^m$); see our work [14] for the detailed study of the convergence of such expressions. $\square$

In our present context, when it comes to obtaining the associated fractal tube zeta function, we cannot, a priori, as in the case of an arbitrary bounded subset of $\mathbb{R}^2$ (see [39], Definition 2.2.8, page 118), directly use an integral formula of the form (for all $s \in \mathbb{C}$ with $\mathcal{R}e(s)$ sufficiently large, and for all $m \in \mathbb{N}^*$ large enough),

$$\tilde{\zeta}_m(s) = \int_0^\eta t^{s-3} \mathcal{V}_m(t) dt = \int_0^\eta t^{s-2} \mathcal{V}_m(t) \frac{dt}{t}, \quad (34)$$

where $\eta > 0$ is chosen sufficiently small, since the tube formulas that we will obtain in Section 4.1 below can only be expressed in an explicit way at a value $\varepsilon_m^m$ of the cohomology infinitesimal. Note that, accordingly, $\tilde{\zeta}_m$ is given by a truncated Mellin transform of the tube function $t \mapsto \mathcal{V}_m(t)$.

In order to bypass this difficulty, we proceed in the same way as in [14]. Given $m \ge 1$ sufficiently large, we set

$$\mathcal{V}_m = \varepsilon^m \mathcal{V}_{partial,m}(\varepsilon) + \frac{\pi \varepsilon^{2m}}{2(N_b - 1)^2}, \quad (35)$$

where
**Proposition 3.3.**

i. For the number of rectangles: $N_b^m = N_b^{x-\{x\}}$.

Due to the fact that

$$ \varepsilon_m^m = \frac{1}{N_b - 1} N_b^{-(\lfloor x \rfloor + \{x\})} = \frac{1}{N_b - 1} N_b^{-x} , \quad N_b^{-x} = (N_b - 1) \varepsilon_m^m , \quad N_b^x = \frac{1}{N_b - 1} (\varepsilon_m^x)^{-1} , $$

one thus also has

$$ N_b^m \varepsilon_m^m = N_b^{x-\{x\}} \frac{1}{N_b - 1} N_b^{-x} = \frac{1}{N_b - 1} N_b^{-\{x\}} . $$

ii. For the elementary horizontal lengths:

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

iii. For the elementary vertical heigths:

$$ h_{j,j+1,m} = \mathcal{O}(L_m^{2-D_W}) = \mathcal{O}(N_b^{(2-D_W)(\{x\}-x)}) = N_b^{(2-D_W)(\{x\}-x)} \mathcal{O}(1), \quad 1 \le j \le N_b^m - 2, $$

where, thanks to inequality (R7) given in Remark 25, page 25, $\mathcal{O}(1)$ may depend on $m$, but is bounded away from 0 and $\infty$; namely, there exist constants independent of $m \in \mathbb{N}$ such that

$$ 0 < C_{inf} \le \mathcal{O}(1) \le C_{sup} < \infty. $$

iv. For the elementary ratios, $0 \le j \le N_b^m - 2$:

$$ \frac{L_m}{h_{j,j+1,m}} = \mathcal{O}(L_m^{D_W-1}) = \mathcal{O}(N_b^{(1-D_W)(\{x\}-x)}) = \mathcal{O}(N_b^{(1-D_W)\{x\}} \varepsilon_m^m) = N_b^{(1-D_W)\{x\}} \varepsilon_m^m \mathcal{O}(1), $$

where, again thanks to inequality (R7) given in Remark 25, page 25, $\mathcal{O}(1)$ may depend on $m$, but is uniformly bounded away from 0 and $\infty$ (i.e, here and in the sequel, independently of $m \in \mathbb{N}^*$ large enough); more specifically,

$$ 0 < \frac{1}{C_{inf}} \le \mathcal{O}(1) \le \frac{1}{C_{sup}} < \infty. $$

**Proposition 3.4 (Basis of the Parallelograms in Common to Overlapping Rectangles).**

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

$$ b_{j-1,j,m} = N_b^{(3D_W-2)\{x\}} (\varepsilon_m^m)^2 \mathcal{O}(1), $$

where,

$$ 0 < C_{inf}^3 \le \mathcal{O}(1) \le C_{sup}^3 < \infty. $$
$$V_{partial,m}(\varepsilon) = \sum_{j=1}^{\#V_m-1} \sum_{k=0}^{m} C_{k,j,m} \varepsilon^{k(2-D_W)} (\varepsilon^{i\ell_{k,j,m}} \mathbf{P} + \varepsilon^{-i\ell_{k,j,m}} \mathbf{P}),$$

that, for the sake of simplicity, we will write in the following form,

$$V_{partial,m}(\varepsilon) = \sum_{j=1}^{\#V_m-1} \sum_{k=0}^{m} \varepsilon^{k(2-D_W)} (\overline{C_{k,j,m}} \varepsilon^{-i\ell_{k,j,m}} \mathbf{P} + C_{k,j,m} \varepsilon^{i\ell_{k,j,m}} \mathbf{P}),$$

where the coefficients $C_{k,j,m}$, given in formula (33), on page 48, denote nonzero complex constants, which depend on $m$, but are uniformly bounded away from 0 and $\infty$ (in $m \in \mathbb{N}^*$ large enough), $\ell_{k,j,m}$ are integers, called the cohomological vertex integers, coming from the Complex Dimensions series expansion of the Weierstrass function (see Theorem 3.2, on page 33 for further details) and where $\varepsilon = \frac{1}{N_b}$ is the intrinsic scale introduced in Definition 3.1, on page 30.

**Definition 4.1** (Natural volume extension – effective distance and tube zeta functions associated to the Weierstrass IFD). Given $m \in \mathbb{N}$ sufficiently large, the $m^{th}$ total effective volume function strict contribution associated with the tubular volume $V_m$ (or two-dimensional Lebesgue measure) of the $\varepsilon_m$-neighborhood of the $m^{th}$ prefractal approximation $\Gamma_m$,

$$\mathcal{D}(\varepsilon_m^m) = \{M = (x, y) \in \mathbb{R}^2, d(M, \Gamma_m) \le \varepsilon_m^m\}, \quad (36)$$

where $\varepsilon_m^m$ is the $m^{th}$ cohomology infinitesimal, as introduced in Definition 3.1, on page 30, is the continuous function given, for all $t \in [0, \varepsilon]$, by substituting the variable $t$ for $\varepsilon$ in $V_{partial,m}(\varepsilon)$ and in $\frac{\pi \varepsilon^{2m}}{2(N_b - 1)^2}$ in relation (35) above (note that we do not substitute $t$ for $\varepsilon$ in $\varepsilon^m = \frac{1}{N_b^m}$, since the expression in relation (35) also represents a mean over the number $N_b^m$ of rectangles involved),

$$\tilde{V}_m(t) = \varepsilon^m V_{partial,m}(t). \quad (37)$$

By construction,

$$\tilde{V}_m(\varepsilon) = V_m + \frac{\pi \varepsilon^{2m}}{2(N_b - 1)^2}.$$

Note that the introduction of this volume function $\tilde{V}_m$ simply amounts to considering an evolving (continuous) tubular neighborhood. Indeed, as was mentioned in the introduction, the knowledge of the expression for the volume at this discrete value is simply the trace, at the value $t = \varepsilon$, of the continuous volume function corresponding to an evolving (continuous) tubular neighborhood; see Fig. 17, on page 51. So, in a sense,
Proof. One has, according to Figure 10, on page 44,

$$ \tan \theta_{j-1,j,m} = \frac{\varepsilon_m^m}{b_{j-1,j,m} + \tilde{b}_{j-1,j,m}}, $$

where $b_{j-1,j,m} + \tilde{b}_{j-1,j,m}$ is the side-length of the parallelogram of basis $\varepsilon_m^m$

$$ \tan (\theta_{j-1,j,m} + \theta_{j,j+1,m}) = \frac{\varepsilon_m^m}{\tilde{b}_{j-1,j,m}}. $$

Hence,

$$ b_{j-1,j,m} + \tilde{b}_{j-1,j,m} = \varepsilon_m^m |\cotan \theta_{j-1,j,m}|, $$

which yields

$$ b_{j-1,j,m} = \varepsilon_m^m |\cotan \theta_{j-1,j,m}| - \tilde{b}_{j-1,j,m} = \varepsilon_m^m \{|\cotan \theta_{j-1,j,m}| - |\cotan (\theta_{j-1,j,m} + \theta_{j,j+1,m})|\}; $$

i.e.,

$$ \begin{align} b_{j-1,j,m} &= \varepsilon_m^m \left( \frac{h_{j-1,j,m}}{L_m} - \left| \cotan \left( \arctan \frac{L_m}{h_{j-1,j,m}} + \arctan \frac{L_m}{h_{j,j+1,m}} \right) \right| \right) \nonumber \\ &= \varepsilon_m^m \left( \frac{h_{j-1,j,m}}{L_m} - \left| \frac{\frac{L_m}{h_{j-1,j,m}} - \frac{L_m}{h_{j,j+1,m}}}{\frac{L_m}{h_{j-1,j,m}} + \frac{L_m}{h_{j,j+1,m}}} \right| \right) \tag{R 16} \\ &= \varepsilon_m^m \left( \frac{h_{j-1,j,m}}{L_m} - \frac{1 - \frac{L_m}{h_{j-1,j,m}} - \frac{L_m}{h_{j,j+1,m}}}{\frac{L_m}{h_{j-1,j,m}} + \frac{L_m}{h_{j,j+1,m}}} \right). \nonumber \end{align} $$

Thanks to Proposition 3.3, on page 50, we have that

$$ \frac{h_{j-1,j,m}}{L_m} = N_b^{(D_W-1)\{x\}} \varepsilon_m^m \mathcal{O}(1) \quad , \text{ with } 0 < C_{inf} \le \mathcal{O}(1) \le C_{sup}. $$

In order to obtain the corresponding estimate for $b_{j-1,j,m}$, we need an asymptotic expansion for $b_{j-1,j,m}$. A slight difficulty occurs, coming from the term

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

The apparent problem is the following:

i. Either one uses, as previously, expressions of the form

$$ \frac{1}{\frac{L_m}{h_{j-1,j,m}} + \frac{L_m}{h_{j,j+1,m}}} = N_b^{(D_W-1)\{x\}} \mathcal{O}(1), $$

with nothing but a black box (which means, unknown terms) in factor of constants, that would yield Complex Dimensions with a real part equal to two, and would therefore lead to a contradiction because the Weierstrass Curve has box dimension $D_W < 2$.

ii. Either, knowing that, which is not the more satisfactorily way of reasoning, from a mathematician's point of view, one copes with it and tries to find how to get rid of those terms.
<figure><img src="image_20.png" /><figcaption>Fig. 17. The evolving tubular neighborhood, for $0 \le t \le \varepsilon_m^m$.</figcaption></figure>

we recover, in an adapted, extended but equivalent manner, the initial theory developed in [39].

As for the resulting $m^{th}$ effective prefractal tube zeta function $\tilde{\zeta}_m^e$ – a generalization to IFDs of the usual definition referred to just above – we define it, for all $s$ in $\mathbb{C}$ with sufficiently large real part (in fact, for $\mathcal{R}e(s) > D_m$, where $D_m$ is the abscissa of convergence of $\tilde{\zeta}_m$), by the following truncated Mellin transform,

$$ \tilde{\zeta}_m^e(s) = \int_0^\varepsilon t^{s-3} \tilde{V}_m(t) dt = \int_0^\varepsilon t^{s-2} \tilde{V}_m(t) \frac{dt}{t}, \qquad (38) $$

where $\varepsilon = \frac{1}{N_b}$ is the intrinsic scale introduced in Definition 3.1, on page 30. (Really, as will be seen in the proof of Theorem 4.2 below, we will drop the term $\frac{\pi t^2}{2(N_b - 1)^2}$ in the definition of $\tilde{V}_m(t)$.)

The choice of the value $\varepsilon$ for the upper bound of the integral in relation (38) (instead of an arbitrary positive number $\eta > 0$ as in the classical theory; see [39], Definition 2.2.8, on page 118) plays an essential role in our present context. Indeed, it corresponds to an intrinsic scale, in connection with the number of divisions (when applying the IFS $\mathcal{T}$). More precisely, the oscillations of the IFD can be characterized by means of (complex powers) of $\varepsilon$, with exponents the underlying Complex Dimensions.

**Remark 4.1.** We stress the fact that $\tilde{\zeta}_m^e$ does not coincide with the usual tube zeta function $\zeta_m$ associated with the $m^{th}$ polygonal prefractal approximation $\Gamma_m \subset \mathbb{R}^2$ to the fractal curve $\Gamma_W$, given, as in [39], for all $s \in \mathbb{C}$ with $\mathcal{R}e(s)$ sufficiently large, by

$$ \tilde{\zeta}_m(s) = \int_0^\varepsilon t^{s-3} V_m(t) dt = \int_0^\varepsilon t^{s-2} V_m(t) \frac{dt}{t}. $$

**Remark 4.2** (Consistency of our approach in the case of the Weierstrass IFD – connection with reality). As shown in Corollary 3.12, on page 46, the $m^{th}$ prefractal approxi-
Two configurations occur:

→ If $h_{j-1,j,m} < h_{j,j+1,m}$, and, thus, $\frac{L_m}{h_{j-1,j,m}} > \frac{L_m}{h_{j,j+1,m}}$, in which case we have that

$$
\begin{align*}
\frac{h_{j-1,j,m}}{L_m} - \frac{1 - \frac{L_m}{h_{j-1,j,m}} \frac{L_m}{h_{j,j+1,m}}}{\frac{L_m}{h_{j-1,j,m}} + \frac{L_m}{h_{j,j+1,m}}} &= \frac{h_{j-1,j,m}}{L_m} - \frac{1 - \frac{L_m}{h_{j-1,j,m}} \frac{L_m}{h_{j,j+1,m}}}{\frac{L_m}{h_{j-1,j,m}} (1 + h_{j-1,j,m} h_{j,j+1,m})} \\
&= \frac{h_{j-1,j,m}}{L_m} - \frac{h_{j-1,j,m}}{L_m} \left(1 - \frac{L_m^2}{h_{j-1,j,m} h_{j,j+1,m}}\right) \left(1 - h_{j-1,j,m} h_{j,j+1,m} + \text{smaller order terms}\right) \\
&= \frac{h_{j-1,j,m}}{L_m} - \frac{h_{j-1,j,m}}{L_m} \left(1 - h_{j-1,j,m} h_{j,j+1,m} - \frac{L_m^2}{h_{j-1,j,m} h_{j,j+1,m}} + L_m^2 + \text{smaller order terms}\right) \\
&= \frac{h_{j-1,j,m}^2 h_{j,j+1,m}}{L_m} + \frac{L_m}{h_{j,j+1,m}} - L_m h_{j-1,j,m} \\
&\qquad + \text{smaller order and negligible terms.}
\end{align*}
$$

Since

$$
\frac{h_{j-1,j,m}^2 h_{j,j+1,m}}{L_m} = N_b^{2(2-D_W)} \{x\} \epsilon_m^m N_b^{(D_W-1)} \{x\} \mathcal{O}(1) = N_b^{(3-D_W)} \{x\} \epsilon_m^m \mathcal{O}(1),
$$

along with

$$
L_m h_{j-1,j,m} = N_b^{(3-D_W)\{x\}} (\epsilon_m^m)^2 \mathcal{O}(1),
$$

and

$$
\frac{L_m}{h_{j,j+1,m}} = N_b^{(1-D_W)\{x\}} (\epsilon_m^m)^2 \mathcal{O}(1) \ll \frac{h_{j-1,j,m}^2 h_{j,j+1,m}}{L_m},
$$

the terms that have to be taken into account in relation (R16), on page 51 above, are then

$$
N_b^{(3-D_W)\{x\}} (\epsilon_m^m)^2 \mathcal{O}(1) = b_{j-1,j,m}.
$$

→ If $h_{j-1,j,m} < h_{j,j+1,m}$, and, thus, $\frac{L_m}{h_{j-1,j,m}} < \frac{L_m}{h_{j,j+1,m}}$, in which case we have that

$$
\frac{h_{j-1,j,m}}{L_m} - \frac{1}{\frac{L_m}{h_{j-1,j,m}} + \frac{L_m}{h_{j,j+1,m}}} = \frac{h_{j-1,j,m}}{L_m} - \frac{h_{j,j+1,m}}{L_m} + \text{smaller order and negligible terms.}
$$

Fortunately, due to results obtained in the proof of Property 2.18, on page 29, this situation
occurs only in the case of reentrant angles, when $N_b \ge 7$, twice, for respectively $[\frac{N_b - 3}{4}]$
consecutive vertices of polygons $\mathcal{P}_{m,k}$, $0 \le k \le N_b^m - 1$. Given a polygon $\mathcal{P}_{m,k}$, and as
already encountered, one just has to reason on the associated first set of consecutive vertices.
The annoying terms simplify two by two in a telescopic sum, from the first reentrant
mations to the Weierstrass Curve become closer and closer to one another and to the Weierstrass fractal Curve, as $m$ increases. Hence, it makes sense to consider a continuous version of the tubular volume, where the discrete and the continuous, in a sense, eventually merge, for all $m \in \mathbb{N}^*$ sufficiently large.

We can also note that, in real life, fractality is not always the result of a discrete process. On the contrary, fractal shapes develop continuously, as is the case, for instance, in biology.

We will show in Section 4.2 below that, in the case of the Weierstrass IFD, and for all integers $m$ sufficiently large, $\tilde{\zeta}_m^e$ has a meromorphic continuation to all of $\mathbb{C}$ and has an abscissa of convergence equal to the Minkowski dimension of $\Gamma_W$ strictly smaller than 2; so that its Complex Dimensions are simple and have real parts strictly smaller than 2. Hence, for any Complex Dimension $\omega$ of the Weierstrass IFD, we have that $\omega$ is simple. (See, especially, Theorem 4.3, on page 54, and Theorem 4.5, on page 56, along with Corollary 4.4, on page 55.)

**Remark 4.3.** We point out that the effective volume function, $\tilde{V}_m$ introduced in Definition 4.1, on page 50, is not only defined for all $t \in [0, \varepsilon]$, but also for all $t \in [0, 1[$. It is also continuous on $[0, 1[$. Indeed, it involves the sum of a locally normally (and hence also, locally uniformly) convergent series of continuous functions (in the variable $t$) on $[0, 1[$. (In fact, for any $0 < \rho < 1$, the general term of the corresponding series can be uniformly bounded, for all $m \in \mathbb{N}^*$ sufficiently large, by the general term of a geometric series with ratio $\rho$.) Naturally, we have that $\tilde{V}_m(0) = 0$.

Since the intrinsic scale $\varepsilon = \frac{1}{N_b}$ belongs to $]0, 1[ = (0, 1)$, this observation justifies, in particular, the fact that the Lebesgue integral initially defining $\tilde{\zeta}_m^e$ in relation (38), on page 51), for $\mathcal{R}e(s) > D_W$, is well defined and convergent, as well as the fact that the effective volume $\tilde{V}_m = \tilde{V}_m(t)$ is a continuous function on $[0, 1[$.

## 4.2. Complex dimensions

We deduce at once the Complex Dimensions of the Weierstrass IFD from the fractal tube formula and the expression for the (local) effective tube zeta function obtained in Theorem 4.1, on page 48, and Theorem 4.2, on page 52 just below, respectively.

**Theorem 4.2 (Effective prefractal tube zeta function for the Weierstrass Iterated Fractal Drums).** For all $m \in \mathbb{N}^*$ sufficiently large, the $m^{th}$ effective prefractal tube zeta function $\tilde{\zeta}_m^e$ of the Weierstrass IFD is well defined (for $\mathcal{R}e(s) > D_W$) and admits a (necessarily unique) meromorphic continuation to all of $\mathbb{C}$, given, for any $s \in \mathbb{C}$, by

$$ \tilde{\zeta}_m^e(s) = \sum_{j=1}^{\#V_m-1} \sum_{k=1}^{m} \varepsilon^{s-D_W+(k-1)(2-D_W)} \frac{\overline{C_{k,j,m}} \varepsilon^{-i \ell_{k,j,m} \mathbf{p}}}{s - D_W + (k-1)(2 - D_W) - i \ell_{k,j,m} \mathbf{p}} $$
vertex, to the penultimate one. There remains the term coming from the first vertex with an interior reentrant angle, that will be denoted $M_{j,m}$, and the term coming from the ultimate one, $M_{j+p-1,m}$: due to the symmetry with respect to the vertical line $x = \frac{1}{2}$ (see Property 2.1, on page 9), they are cancelled by those coming from the symmetric polygon, see Figure 17, on page 54). To summarize, one obtains a sum of the form

$$ \frac{h_{j-1,j,m}}{L_m} - \frac{h_{j,j+1,m}}{L_m} + \frac{h_{j,j+1,m}}{L_m} - \frac{h_{j+1,j+2,m}}{L_m} + \frac{h_{j+1,j+2,m}}{L_m} \dots - \frac{h_{j+p,j+p+1,m}}{L_m} \underbrace{\quad telescoping sum \quad}_{\text{telescoping sum}} $$

The remaining terms $\frac{h_{j-1,j,m}}{L_m}$ and $-\frac{h_{j+p,j+p+1,m}}{L_m}$ are the ones which will simplify with the exact opposites coming from the symmetric polygon with respect to the vertical line $x = \frac{1}{2}$ (see Figure 17, on page 54), since

$$ \begin{align*} \frac{h_{j+p,j+p+1,m}}{L_m} &= \frac{1}{L_m} \left| W\left(\frac{j+p+1}{(N_b-1)N_b^m}\right) - W\left(\frac{j+p}{(N_b-1)N_b^m}\right) \right| \\ &= \frac{1}{L_m} \left| W\left(\frac{(N_b-1)N_b^m - j - p - 1}{(N_b-1)N_b^m}\right) - W\left(\frac{(N_b-1)N_b^m j - p}{(N_b-1)N_b^m}\right) \right| \\ &= \frac{h_{(N_b-1)N_b^m - j - p - 1, (N_b-1)N_b^m - j - p, m}}{L_m}. \end{align*} $$

Thus, in the end, there is no problem.

In the light of the above results, one may now rewrite $b_{j-1,j,m}$ as follows:

$$ b_{j-1,j,m} = N_b^{(3-D_W)\{x\}} (\varepsilon_m^m)^2 \mathcal{O}(1), \qquad (\mathcal{R} 17) $$

where, thanks to inequality (R7) given in Remark 25, on page 25,

$$ 0 < C_{inf}^3 \le \mathcal{O}(1) \le C_{sup}^3 < \infty. $$

This concludes the proof of Proposition 3.4, stated on page 50.

<figure><img src="image_18.png" /></figure>
$$ + \sum_{j=1}^{\#V_m-1} \sum_{k=1}^{m} \varepsilon^{s-D_W+(k-1)(2-D_W)} \frac{C_{k,j,m} \varepsilon^{i \ell_{k,j,m} \mathbf{P}}}{s - D_W + (k-1)(2-D_W) + i \ell_{k,j,m} \mathbf{P}}, \quad (39) $$

where, for $1 \le j \le \#V_m - 1$, $1 \le k \le m$, the coefficients $C_{k,j,m}$, given in formula (33), on page 48, denote nonzero complex constants, which depend on $m$, but are uniformly bounded away from 0 and $\infty$ (in $m \in \mathbb{N}^*$ large enough); $\ell_{j_{k,m},k} \in \mathbb{N}$ is the cohomological vertex integer associated with the vertex $M_{j_{k,m},k}$ (see Definition 3.2, on page 32).

Furthermore, still for all $m \in \mathbb{N}^*$ sufficiently large, $\tilde{\zeta}_m^e$ admits a well-defined abscissa of convergence, given by

$$ D_W = 2 + \frac{\ln \lambda}{\ln b} = 2 - \ln b \frac{1}{\lambda}. $$

Note that $D_W \in (1,2)$. In particular, $\tilde{\zeta}_m^e$ is holomorphic for all $s \in \mathbb{C}$ with $\mathrm{Re}(s) > D_W$.

Moreover, the open half-plane $\{s \in \mathbb{C} : \mathrm{Re}(s) > D_W\}$ is the half-plane of holomorphic continuation (resp., of convergence) of $\tilde{\zeta}_m^e$, in the sense of [39]; i.e., it is the largest open half-plane on which $\tilde{\zeta}_m^e$ is holomorphic (resp., on which the Lebesgue integral initially defining $\tilde{\zeta}_m^e(s)$ in relation (38), on page 51), is convergent).

As is proved in [14], $\tilde{\zeta}_m^e$, the $m^{th}$ local tube zeta function of the Weierstrass IFD, is also the contribution of the $m^{th}$ prefractal approximation $\Gamma_m$ to $\tilde{\zeta}_{\Gamma_W}^e$, the effective global tube zeta function of the Weierstrass IFD to be given in Theorem 4.3 below, on page 54.

**Proof.** In essence, in light of Theorem 4.1, on page 48, the main statement of the theorem follows from a direct computation based on relation (31), on page 48, in Definition 4.1, on page 50, combined with the local normal (and hence also, uniform) convergence on all of $\mathbb{C}$ (and in the variable $s \in \mathbb{C}$) of the series of functions appearing on the right-hand side $\Sigma(s)$ of relation (39), on page 53.

Note that we exclude here the (artificial) terms in $\frac{\varepsilon^s}{s}$, coming from the extreme wedges, and from the restriction of our study to $[0, 1[$.

The aforementioned direct computation (used in conjunction with Theorem 4.1) for all $m \in \mathbb{N}^*$ large enough and all $s \in \mathbb{C}$ with $\mathrm{Re}(s) > D_W$, and Weierstrass's Theorem (for series of locally uniformly convergent holomorphic functions) yields the holomorphicity of $\tilde{\zeta}_m^e$ for $\mathrm{Re}(s) > D_W$.

Using again the local uniform convergence of the series of meromorphic functions appearing on the right-hand side of relation (39) and (e.g., as in [44], Appendix C, and [39]) viewing meromorphic functions as continuous functions with values on the Riemann sphere (equipped with the chordal metric), we conclude that the Principle of Analytic (i.e., Meromorphic, here) Continuation (and since $\tilde{\zeta}_m^e(s) = \Sigma(s)$ for all $s \in \mathbb{C}$ with $\mathrm{Re}(s) > D_W$) implies that, still for all $m \in \mathbb{N}^*$ sufficiently large, $\tilde{\zeta}_m^e$ is meromorphic in all of $\mathbb{C}$ and coincides with $\Sigma$; i.e., relation (39) holds for all $s \in \mathbb{C}$, as required.
<figure><img src="image_19.png" /><figcaption>Figure 17: The symmetric points with respect to the vertical line $x = \frac{1}{2}$, leading to terms that cancel each other out in the proof of Proposition 3.4.</figcaption></figure>

In the sequel, we will use the following two power series expansions:

$$i. \forall z \in [0, 1[: \sqrt{1+z} = \sum_{k=0}^{\infty} \binom{\frac{1}{2}}{k} z^k,$$

where, for any pair of natural integer $k$, $\binom{\frac{1}{2}}{k}$ is the generalized binomial coefficient

$$\binom{\frac{1}{2}}{k} = \frac{\frac{1}{2} \times (\frac{1}{2} - 1) \times (\frac{1}{2} - 2) \times \cdots \times (\frac{1}{2} - k + 1)}{k!} = \frac{\binom{\frac{1}{2}}{k}}{k!}.$$

This expansion is thus valid for

$$z = \frac{L_m^2}{h_{j-1,j,m}^2} = O\left(L_m^{2(D_W-1)}\right) \ll 1.$$

$$ii. \forall z \in [0, 1[: \tan^{-1} z = \arctan z = \sum_{k=0}^{\infty} \frac{(-1)^k z^{2k+1}}{2k+1}, \text{ which is also valid for}$$

$$z = \frac{L_m^2}{h_{j-1,j,m}^2} = O\left(L_m^{2(D_W-1)}\right) \ll 1.$$
We note for later use (in the proof of Corollary 4.4, on page 55) that the same proof shows that the set of possible poles of $\tilde{\zeta}_m^e$ (in $\mathbb{C}$) is given by

$$ \mathcal{D}_m = \mathcal{D}_m^- \cup \mathcal{D}_m^+, \qquad (40) $$

where

$$ \mathcal{D}_m^- = \{D_W - k(2 - D_W) - i \ell_{k,j,m} \mathbf{p}, \text{ with } 0 \le k \le m, 0 \le j \le \#V_m - 1\} \quad (41) $$

and

$$ \mathcal{D}_m^+ = \{D_W - k(2 - D_W) + i \ell_{k,j,m} \mathbf{p}, \text{ with } 0 \le k \le m, 0 \le j \le \#V_m - 1\}. \quad (42) $$

In addition, there exists $m_0 \in \mathbb{N}^*$ such that for any $\omega \in \mathcal{D}_m$, $\omega$ is an actual and simple Complex Dimension of $\Gamma_m$, for all $m \ge m_0$; i.e., $\mathrm{res}(\tilde{\zeta}_m^e, \omega) \ne 0$. This last statement follows from part (i.) of Remark 4.4 below, on page 54.

Finally, the fact that for any $\delta > 0$, $\tilde{\zeta}_m^e$ cannot be holomorphic on the open half-plane $\{s \in \mathbb{C} : \mathrm{Re}(s) < D_W + \delta\}$ now follows, for example, from the fact that $D_W$ is an actual pole of $\tilde{\zeta}_m^e$. $\square$

**Theorem 4.3** (Effective global tube zeta function for the Weierstrass Iterated Fractal Drums). By analogy with the work in [39], we define the effective global tube zeta function $\tilde{\zeta}_{\Gamma_W}^e$ of the Weierstrass IFD, by the following expression, for all $s \in \mathbb{C}$,

$$ \tilde{\zeta}_{\Gamma_W}^e(s) = \lim_{m \to \infty} \tilde{\zeta}_m^e(s). \qquad (43) $$

$\tilde{\zeta}_{\Gamma_W}^e$ admits a (necessarily unique) meromorphic continuation to all of $\mathbb{C}$ (see [14] for the proof of the existence of the limit, which is locally uniform on $\mathbb{C}$).

The (technical) proof of most of Theorem 4.3 is given in Appendix A.6, on page 74.

**Remark 4.4.** i. Let $m \in \mathbb{N}^*$ be arbitrary, but sufficiently large, and $\omega$ be a potential pole (necessarily simple) of $\tilde{\zeta}_m^e$ – or, equivalently, of $\tilde{\zeta}_m^e$ (since $D_W < 2$); $\omega$ is a possible Complex Dimension of the Weierstrass IFD, as given in Theorem 4.5 below, on page 56. Then, we claim that $\omega$ is an *exact* and *simple* Complex Dimension of $\Gamma_m$.

Indeed, say that

$$ \omega = \omega_{k,j,m} = D_W - k(2 - D_W) + i \ell_{k,j,m} \mathbf{p}, \qquad (44) $$

for some $m \in \mathbb{N}^*$ sufficiently large and $0 \le k \le m$. Then, according to Theorem 4.3, given on page 54, we have that

$$ \mathrm{res} \left( \tilde{\zeta}_m^e, \omega_{k,j,m} \right) = \lim_{s \to \omega_{k,j,m}} (s - \omega_{k,j,m}) \tilde{\zeta}_m^e(s) = C_{k,j,m} \neq 0. \qquad (45) $$
**Notation 10.** In the sequel, for the sake of simplicity, we will use the following notation:

i. $\sum_{j \text{ rectangle}} \dots$, to denote a sum involving all the upper and lower rectangles, which amounts to taking into accounts indices $j$ such that $1 \le j \le (N_b - 1) N_b^m$.

ii. $\sum_{j \text{ lower wedge}} \dots$, to denote a sum involving all the lower wedges, which amounts to taking into accounts indices $j$ such that $N_b^m (N_b - 2 \lfloor \frac{N_b - 3}{4} \rfloor) - 1$.

iii. $\sum_{j \text{ upper wedge}} \dots$, to denote a sum involving all the upper wedges, which amounts to taking into accounts indices $j$ such that $N_b^m (1 + 2 \lfloor \frac{N_b - 3}{4} \rfloor) - 1$.

And, similarly:

iv. $\sum_{j \text{ upper triangle}} \dots$, to denote a sum involving all the extra outer upper triangles.

v. $\sum_{j \text{ lower triangle}} \dots$, to denote a sum involving all the extra outer lower triangles.

vi. $\sum_{j \text{ lower parallelogram}} \dots$, to denote a sum involving all the upper overlapping rectangles.

vii. $\sum_{j \text{ upper parallelogram}} \dots$, to denote a sum involving all the lower overlapping rectangles.
Therefore, we see that $\omega = \omega_{k,j,m}$ is an actual pole (and necessarily a simple pole) of $\zeta_m^e$.

ii. Moreover, also in agreement with the higher-dimensional theory developed in [39] (see also [38] and [41], for example), the Complex Dimensions of the Weierstrass IFD can be defined indifferently via the $m^{th}$ effective prefractal tube zeta functions $\tilde{\zeta}_m^e$, for all $m \in \mathbb{N}^*$ sufficiently large (i.e., for all $m \ge m_0$, with $m_0 \in \mathbb{N}^*$ optimal).

iii. Parts i. and ii. of this remark are valid, in an obviously adapted form, both for the potential (or possible) Complex Dimensions and for the exact Complex Dimensions of the Weierstrass IFD.

**Corollary 4.4** (Complex dimensions of the prefractal approximations). For all $m \in \mathbb{N}^*$ sufficiently large, the set $\mathbb{D}_m$ of possible Complex Dimensions of the prefractal approximations $\Gamma_m$, obtained as poles of the $m^{th}$ effective prefractal tube zeta function is given by

$$ \mathbb{D}_m = \mathbb{D}_m^- \cup \mathbb{D}_m^+, \qquad (46) $$

where

$$ \mathbb{D}_m^- = \{D_W - k(2 - D_W) - i \ell_{k,j,m} \mathbf{p}, \quad \text{with } 0 \le k \le m, 0 \le j \le \#V_m - 1\} \quad (47) $$

and

$$ \mathbb{D}_m^+ = \{D_W - k(2 - D_W) + i \ell_{k,j,m} \mathbf{p}, \quad \text{with } 0 \le k \le m, 0 \le j \le \#V_m - 1\}. \quad (48) $$

Here, the positive integers $\ell_{k,j,m}$ are the cohomological vertex integers given in Corollary 3.3, on page 36, and $\mathbf{p} = \frac{2\pi}{\ln N_b}$ is the oscillatory period of the Weierstrass IFD.

Furthermore, still for all $m \in \mathbb{N}^*$ sufficiently large, all of the possible Complex Dimensions of $\Gamma_m$ are exact (i.e., actual poles of $\zeta_m^e$ or $\tilde{\zeta}_m^e$) and simple.

**Proof.** This corollary follows at once from Theorem 4.2, on page 52, combined with parts i. and ii. of Remark 4.4, on page 54.

For all $m \in \mathbb{N}^*$ sufficiently large (i.e., for all $m \ge m_0$, with $m_0 \in \mathbb{N}^*$ optimal), we have the strict inclusion

$$ \mathbb{D}_m \subset \mathbb{D}_{m+1}. $$

As a result, the prefractal approximations $\Gamma_m$ (for all $m \ge m_0$) are not only fractal (i.e., have nonreal Complex Dimensions) in the sense of [42], [44], [39], [29], but are also fractal in increasingly many dimensions $d_k = D_W - k(2 - D_W)$, for $0 \le k \le m$ (i.e., have nonreal Complex Dimensions with real part $d_k$, in the sense of [39], [29]). Note that $d_m = D_W - m(2 - D_W) \to -\infty$ as $m \to \infty$. This is natural and important from
**Proposition 3.5 (Contribution of the Rectangles to the Tubular Volume).**

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

$$
\begin{align*}
\mathcal{V}_{m, \Gamma_{W_m}, \text{Rectangles}}(\varepsilon_m^m) &= 2 \sum_{j \text{ rectangle}} \varepsilon_m^m \ell_{j-1,j,m} \\
&= 2 \sum_{j \text{ rectangle}} \varepsilon_m^m \sqrt{L_m^2 + h_{j-1,j,m}^2} \\
&= 2 \sum_{j \text{ rectangle}} \varepsilon_m^m h_{j-1,j,m} \sqrt{1 + \frac{L_m^2}{h_{j-1,j,m}^2}} \\
&= 2 \sum_{j \text{ rectangle}} \varepsilon_m^m h_{j-1,j,m} \sqrt{1 + \frac{L_m^2}{h_{j-1,j,m}^2}} \\
&= 2 \sum_{j \text{ rectangle}} \varepsilon_m^m h_{j-1,j,m} \sum_{k=0}^{\infty} \left(\frac{1}{k}\right)^{\frac{1}{2}} \frac{L_m^{2k}}{h_{j-1,j,m}^{2k}} \\
&= 2 \sum_{j \text{ rectangle}} \varepsilon_m^m h_{j-1,j,m} \sum_{k=0}^{\infty} \left(\frac{1}{k}\right)^{\frac{1}{2}} N_b^{k(2-D_W)} \{x\} (\varepsilon_m^m)^{k(2-D_W)} \mathcal{O}(1) \\
&= 2 \sum_{j \text{ rectangle}} \varepsilon_m^m (\varepsilon_m^m)^{2-D_W} \mathcal{O}(1) \sum_{k=0}^{\infty} \left(\frac{1}{k}\right)^{\frac{1}{2}} N_b^{k(2-D_W)} \{x\} (\varepsilon_m^m)^{k(2-D_W)} \mathcal{O}(1) \\
&= 2 N_b^m (\varepsilon_m^m)^{2-D_W} \mathcal{O}(1) \sum_{k=0}^{\infty} \left(\frac{1}{k}\right)^{\frac{1}{2}} N_b^{k(2-D_W)} \{x\} (\varepsilon_m^m)^{1+k(2-D_W)} \mathcal{O}(1) \\
&= 2 (N_b - 1) N_b^m \varepsilon_m^m (\varepsilon_m^m)^{2-D_W} \sum_{k=0}^{\infty} \left(\frac{1}{k}\right)^{\frac{1}{2}} N_b^{k(2-D_W)} \{x\} (\varepsilon_m^m)^{k(2-D_W)} \mathcal{O}(1) \\
&= 2 N_b^{-\{x\}} (\varepsilon_m^m)^{2-D_W} \sum_{k=0}^{\infty} \left(\frac{1}{k}\right)^{\frac{1}{2}} N_b^{k(2-D_W)} \{x\} (\varepsilon_m^m)^{k(2-D_W)} \mathcal{O}(1) \\
&= 2 \sum_{k=0}^{\infty} \left(\frac{1}{k}\right)^{\frac{1}{2}} N_b^{(k(2-D_W)-1)\{x\}} (\varepsilon_m^m)^{2-D_W+k(2-D_W)} \mathcal{O}(1), \tag{R 18}
\end{align*}
$$

where, for notational simplicity, we have used the estimates obtained in relation (R9), given on page 27,
for the elementary quotients $\frac{L_m}{h_{j-1,j,m}}$, in the form

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

where $\mathcal{O}(1)$ may depend on $m$, but is uniformly bounded away from $0$ and $\infty$; more specifically,

$0 < C_{inf} \le \mathcal{O}(1) \le C_{sup} < \infty .$

This ensures here that, for all $k \in \mathbb{N}$,

$$
2 \binom{\frac{1}{2}}{k} N_b^{(k(2-D_W)-1)\{x\}} (\varepsilon_m^m)^{2-D_W+k(2-D_W)} \mathcal{O}(1) > 0. \qquad (\mathbb{R} 19)
$$

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.
a physical point of view since, clearly, ideal mathematical fractals do not exist in nature, but are increasingly well approximated by fractal-like objects (here, their natural prefractal approximations). □

**Remark 4.5.** Observe that the reason why we had to introduce the notion of effective volume $\tilde{V}_m$ (and hence also, the corresponding notion of effective local fractal zeta functions $\tilde{\zeta}_m^e$ and $\zeta_m^e$) is that it is not possible for $\tilde{V}_m(t)$ to coincide with the area of the t-neighborhood, for all $t \in [0, \varepsilon]$ (or even for all $t \ge 0$ sufficiently small) because then, $\tilde{\zeta}_m^e$ would coincide with the usual tube zeta function $\zeta_m^e$ from [39] (and hence, by [39], Theorem 2.2.11) the abscissa of convergence $D_W$ would have to be equal to 1 (the Minkowski dimension of the $m^{th}$ polygonal prefractal approximation $\Gamma_m$): in short, $D_W = 1$, which is a contradiction since $D_W > 1$.

**Theorem 4.5 (Complex dimensions of the Weierstrass IFD).** The Complex Dimensions of the Weierstrass IFD $\Gamma_W^I$, obtained as poles of the $m^{th}$ tube (or, equivalently, of the distance) effective zeta functions, are all exact and simple, and the resulting set $\mathbb{D}$ of the Complex Dimensions of $\Gamma_W^I$ is given as follows:

$$ \mathbb{D} = \bigcup_{m=0}^{\infty} (\mathbb{D}_{-m} \cup \mathbb{D}_{+m}), \qquad (49) $$

where

$$ \mathbb{D}_{-m} = \{D_W - k(2 - D_W) - i \ell_{k,j,m} \mathbf{p}, \quad \text{with } 0 \le k \le m, 0 \le j \le \#V_m - 1\}, \quad (50) $$

and

$$ \mathbb{D}_{+m} = \{D_W - k(2 - D_W) + i \ell_{k,j,m} \mathbf{p}, \quad \text{with } 0 \le k \le m, 0 \le j \le \#V_m - 1\}, \quad (51) $$

where $m \in \mathbb{N}^*$ sufficiently large is arbitrary (i.e., for all $m \ge m_0$, for some optimal $m_0 \in \mathbb{N}^*$). Here, the positive integers $\ell_{k,j,m}$ are given in Corollary 3.3, on page 36, and $\mathbf{p} = \frac{2\pi}{\ln N_b}$ is the oscillatory period of the Weierstrass IFD.

In other words (see Remark 4.6 just below, on page 57),

$$ \mathbb{D} = \{\omega_{k,j,m} = D_W - k(2 - D_W) + i \ell \mathbf{p}, \quad \text{with } k \in \mathbb{N} \text{ and } \ell \in \mathbb{Z} \text{ arbitrary}\}. \qquad (52) $$

Accordingly, as is described in Fig. 18 and its caption, on page 57, the Complex Dimensions of $\Gamma_W^I$ are periodically distributed (with period $\mathbf{p} = \frac{2\pi}{\ln N_b}$) along countably many vertical lines, with abscissae $D_W - k(2 - D_W)$, where $k \in \mathbb{N}$ is arbitrary.

Consequently, the Weierstrass IFD $\Gamma_W^I$ is fractal, in the sense of the theory of Complex Dimensions developed in [42], [43], [44], [39] and [29] (i.e., has nonreal Complex Dimensions).
**Proposition 3.6 (Contribution of the Extreme, Upper and Lower Wedges to the Tubular Volume).**

i. Given $m \in \mathbb{N}^*$ sufficiently large, the contribution of the **extreme wedges** to the tubular volume is given by

$$ \mathcal{V}_{m, \Gamma_{W_m}, \text{extreme wedges}}(\varepsilon_m^m) = \pi (\varepsilon_m^m)^2 . $$

ii. Given $m \in \mathbb{N}^*$ sufficiently large, the contribution of the $r_b^+ N_b^m - 1$ **upper wedges** to the tubular volume is given by

$$
\begin{align*}
\mathcal{V}_{m, \Gamma_{W_m}, \text{upper wedges}} (\varepsilon_m^m) &= \frac{1}{2} \sum_{j \text{ upper wedge}} (\pi - \theta_{j-1,m} - \theta_{j,j+1,m}) (\varepsilon_m^m)^2 \\
&= \frac{1}{2} \sum_{j \text{ upper wedge}} (\varepsilon_m^m)^2 \left( \pi - \arctan \frac{L_m}{h_{j-1,j,m}} - \arctan \frac{L_m}{h_{j,j+1,m}} \right) \\
&= \frac{(\varepsilon_m^m)^2}{2} \sum_{j \text{ upper wedge}} (\varepsilon_m^m)^2 \left( \pi - \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \frac{L_m^{2k+1}}{h_{j-1,j,m}^{2k+1}} - \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \frac{L_m^{2k+1}}{h_{j,j+1,m}^{2k+1}} \right) \\
&= \frac{(\varepsilon_m^m)^2}{2} \sum_{j \text{ upper wedge}} \left( \pi - \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} N_b^{(2k+1)(1-D_W)\{x\}} (\varepsilon_m^m)^{2k+1} \mathcal{O}(1) \right) \\
&= \frac{(\varepsilon_m^m)^2}{2} (r_b^+ N_b^m - 1) \left( \pi - \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} N_b^{(2k+1)(1-D_W)\{x\}} (\varepsilon_m^m)^{2k+1} \mathcal{O}(1) \right) \\
&= \frac{(\varepsilon_m^m)^2}{2} \left( \frac{\varepsilon_m^m}{4} r_b^+ N_b^{-(x)} - \frac{(\varepsilon_m^m)^2}{2} \right) \left( \pi - \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} N_b^{((2k+1)(1-D_W)\{x\})} (\varepsilon_m^m)^{2k+1} \mathcal{O}(1) \right) \\
&= \frac{\pi}{2} \left( \frac{(\varepsilon_m^m)^3}{4} r_b^+ N_b^{-(x)} - \frac{(\varepsilon_m^m)^4}{2} \right) - \frac{(\varepsilon_m^m)^3}{4} r_b^+ \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} N_b^{-(((2k+1)D_W-2k)\{x\})} (\varepsilon_m^m)^{2k+1} \mathcal{O}(1) \\
&\qquad + \frac{(\varepsilon_m^m)^4}{4} \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} N_b^{-(((2k+1)D_W-2k+1)\{x\})} (\varepsilon_m^m)^{2k+1} \mathcal{O}(1), \tag{R 20}
\end{align*}
$$

where, for notational simplicity, and as done previously in Proposition 3.5, on page 56, we have used the estimates obtained in relation (R9), given on page 27, for the elementary quotients $\frac{L_m}{h_{j-1,j,m}}$, in the form

$$ \frac{h_{j-1,j,m}}{L_m} = L_m^{1-D_W} \mathcal{O}(1) , \quad \frac{h_{j,j+1,m}}{L_m} = L_m^{1-D_W} \mathcal{O}(1) , $$

where, as in Proposition 3.5, on page 56 above, $\mathcal{O}(1)$ may depend on $m$, but is uniformly bounded away from 0 and $\infty$; more specifically,

$$ 0 < C_{inf} \le \mathcal{O}(1) \le C_{sup} < \infty . $$

This ensures here that, for all $k \in \mathbb{N}$,

$$ \frac{(-1)^k}{2k+1} N_b^{-(((2k+1)D_W-2k+1)\{x\})} (\varepsilon_m^m)^{2k+1} \mathcal{O}(1) \ne 0, \qquad (\mathbb{R} 21) $$
<figure><img src="image_21.png" /><figcaption>**Fig. 18.** The Complex Dimensions of the Weierstrass IFD. The nonzero Complex Dimensions are periodically distributed (with the same period $\mathbf{p} = \frac{2\pi}{\ln N_b}$, the oscillatory period of the Weierstrass IFD) along countably many vertical lines, with abscissae $D_W - k(2 - D_W)$, where $k \in \mathbb{N}$ is arbitrary. For the sake of representation, there is a different color for each vertical line, and a specific symbol is used to plot the imaginary parts of the Complex Dimensions associated with a given vertical line.</figcaption></figure>

Moreover, $\Gamma_W^I$ is fractal in infinitely many dimensions, in the sense of [39], [29]. More specifically, it is fractal in the countably infinite set of dimensions $d_k = D_W - k(2 - D_W)$, where $k \in \mathbb{N}$ is arbitrary, with $d_k \to -\infty$ as $k \to \infty$.

**Remark 4.6.** In light of relation (49), relation (52) holds because – as follows from Definition 3.2, on page 32, the cohomological vertex integers $\ell_{k,j,m}$ describe all of $\mathbb{N}$ as $m \to \infty$ and for $0 \le k \le m$ (since these integers only label the points of the prefractal approximations $V_m$).

### 4.2.1. Possible interpretation

Fig. 18, on this page, depicts the distribution of the Complex Dimensions of the Weierstrass IFD $\Gamma_W^I$, as given by Theorem 4.5, on page 56. In order to understand their deeper meaning, one may consider an horizontal (and discrete) $\ell \mathbf{p}$ line, of equation $y = \ell \mathbf{p}$, where $\ell \in \mathbb{Z}$ is arbitrary (but fixed). Such a line corresponds to the $\ell^{th}$ order vibration mode, but which can also be interpreted as coming from:

i. The vertical line $x = D_W$, which, this time, corresponds to oscillations coming from the whole prefractal $\Gamma_m$ itself.

ii. The vertical lines $x = D_W - k(2 - D_W)$, with $k$ in $\mathbb{N}^* = \mathbb{N} \setminus \{0\}$.
iii. In the same way, given $m \in \mathbb{N}^*$ sufficiently large, the contribution of the $r_b^- N_b^m - 1$ lower wedges to the tubular volume is given by

$$ 
\begin{aligned}
V_{m, \Gamma_{W_m}, \text{lower wedges}} (\varepsilon_m^m) = & \frac{\pi}{2} \left( \frac{(\varepsilon_m^m)^3}{4} r_b^- N_b^{-\{x\}} - \frac{(\varepsilon_m^m)^4}{2} \right) \\
& - \frac{(\varepsilon_m^m)^3}{4} r_b^- \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} N_b^{-((2k+1)D_W-2k)\{x\}} (\varepsilon_m^m)^{2k+1} \mathcal{O}(1) \\
& + \frac{(\varepsilon_m^m)^4}{4} \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} N_b^{-((2k+1)D_W-2k+1)\{x\}} (\varepsilon_m^m)^{2k+1} (\varepsilon_m^m)^{2k+1} \mathcal{O}(1).
\end{aligned}
\tag{R 22} $$

As previously, we obtain that, for all $k \in \mathbb{N}$,

$$ \frac{(-1)^k}{2k+1} N_b^{-((2k+1)D_W-2k+1)\{x\}} (\varepsilon_m^m)^{2k+1} \mathcal{O}(1) \neq 0. \tag{R 23} $$

**Proposition 3.7 (Negative Contribution of the Extra Outer Triangles to the Tubular Volume).**

i. Given $m \in \mathbb{N}^*$ sufficiently large, the negative contribution of the $(N_b - r_b^+ - 1) N_b^m$ extra outer lower triangles to the tubular volume is given by

$$ 
\begin{aligned}
V_{m, \Gamma_{W_m}, \text{extra outer lower triangles}} (\varepsilon_m^m) &= -\frac{\varepsilon_m^m}{2} \sum_{j \text{ triangle}} \{b_{j-1,j,m} + b_{j,j+1,m}\} \\
&= -\frac{\varepsilon_m^m}{2} \sum_{j \text{ lower triangle}} N_b^{(3-D_W)\{x\}} (\varepsilon_m^m)^2 \mathcal{O}(1) \\
&= -\frac{\varepsilon_m^m}{2} (N_b - r_b^+ - 1) N_b^m N_b^{(3-D_W)\{x\}} (\varepsilon_m^m)^2 \mathcal{O}(1) \\
&= -\frac{(\varepsilon_m^m)^2}{2} (N_b - r_b^+ - 1) N_b^{-\{x\}} N_b^{(3-D_W)\{x\}} \mathcal{O}(1),
\end{aligned}
\tag{R 24} $$

where the coefficient $r_b^+$ is defined in formula (R14) page 41, and where, as in Proposition 3.5, on page 56 above, $\mathcal{O}(1)$ may depend on $m$, but is uniformly bounded away from 0 and $\infty$; more specifically,

$$ 0 < C_{inf}^3 \le \mathcal{O}(1) \le C_{sup}^3 < \infty, $$

thanks to the estimates obtained in Corollary 3.4, on page 50, and where the coefficient $r_b^+$ is defined in formula (R14), on page 41.

This ensures here that, for all $k \in \mathbb{N}$,

$$ \frac{(-1)^k}{2k+1} N_b^{-(2k+1)(D_W-1)\{x\}} (\varepsilon_m^m)^{2k+1} \mathcal{O}(1) \neq 0. \tag{R 25} $$
For $k \le m$, it corresponds to oscillations coming from the prefralctal graphs $\Gamma_{m-k}$, a phenomenon which can be understood via the following consideration:

Switching from the $(m-k)^{th}$ prefractal graph, to the $m^{th}$ one, $0 < k \le m$, is done by applying $k$ iterates of the $T_j$ maps (as defined in Proposition 2.2, on page 10),

$$T_{j_1 \dots j_k} = T_{j_1} \circ \dots \circ T_{j_k}. \qquad (53)$$

with $j_1, \dots, j_k \in \{0, \dots, N_b - 1\}$.

In terms of the vertical distance between consecutive vertices, this amounts to a multiplication of the amplitudes of the vibrations by the factor $\lambda^k = N_b^{-k(2-D_W)}$, associated to a sum of cosine expressions.

It thus provides an interesting interpretation of the real parts

$$D_W - k(2 - D_W), \quad \text{for } 0 < k \le m, \qquad (54)$$

insofar as the $m^{th}$ prefractal graph bears – or, in a sense, *feels* – the oscillations of its predecessors.

There remains the lines $x = D_W - k(2 - D_W)$, with $k > m$.

In order to interpret them, one could think in the same way, but, without associated graphs, how? Except if they could exist, in some way. This could be the purpose of a later extension of the prefractal sequence $(\Gamma_m)_{m \in \mathbb{N}}$, a priori indexed by nonnegative integers, to negative ones, via the new concept of *antefractals*. However, this issue will not be further discussed in the present paper.

### 4.2.2. Analogy with the general theory of complex dimensions

Our results in Theorem 4.1, on page 48, on the fractal tube formula for the Weierstrass IFD, are similar to the general (exact, pointwise) fractal tube formulas (via tube zeta functions) obtained in the higher-dimensional theory of Complex Dimensions in [39] (Chapter 5), or in [41], and extending the fractal tube formulas for fractal strings obtained in [42], [43] and [44] (Chapter 8). Compare, e.g., in the case of simple poles and under the hypothesis of strong languidity (a strong form of polynomial growth condition) of $\tilde{\zeta}_m^e$ [39], Theorem 5.1.16, page 427.

There is a notable difference, however, due to the great complexity of the Weierstrass Curve $\Gamma_W$ and of the associated IFD $\Gamma_W^T$. Namely, the fractal tube formula is only given for the volume $V_m$ of the $m^{th}$ prefractal approximation $\Gamma_m$, and evaluated at the $m^{th}$ cohomology infinitesimal $\varepsilon_m^m$, for all sufficiently large $m \in \mathbb{N}$.

Indeed, according to the aforementioned results from [39] and [41], we would have, in particular, that the tubular volume is given as follows:

$$V_m = \sum_{\omega} \mathrm{res} \left( \tilde{\zeta}_{m, \omega}^e \right) \varepsilon^{2-\omega} = \sum_{\omega} \frac{\mathrm{res} \left( \zeta_{m, \omega}^e \right)}{2-\omega} \varepsilon^{2-\omega}, \qquad (55)$$
ii. In the same way, given $m \in \mathbb{N}^*$ sufficiently large, the negative contribution of the $(N_b - r_b^- - 1) N_b^m$ extra outer upper triangles to the tubular volume is given by

$$ V_{m, \Gamma_{W_m}, \text{extra outer upper triangles}} (\varepsilon_m^m) = - \frac{(\varepsilon_m^m)^2}{2} (N_b - r_b^- - 1) N_b^{-\{x\}} N_b^{(3-D_W)\{x\}} \mathcal{O}(1), \qquad (\mathbb{R} \ 26) $$

again where, as in Proposition 3.5, on page 56 above, $\mathcal{O}(1)$ may depend on $m$, but is uniformly bounded away from 0 and $\infty$; more specifically,

$$ 0 < C_{inf}^3 \le \mathcal{O}(1) \le C_{sup}^3 < \infty, $$

and where the coefficient $r_b^-$ is defined in formula (R15), on page 41.

**Proposition 3.8 (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

$$ V_{m, \Gamma_{W_m}, \text{upper and lower parallelograms}} (\varepsilon_m^m) = - \varepsilon_m^m \sum_{j \text{ upper and lower parallelogram}} b_{j-1,j,m} = - (\varepsilon_m^m)^2 N_b^{(3-D_W)\{x\}} \mathcal{O}(1), \qquad (\mathbb{R} \ 27) $$

where, as in Proposition 3.5, on page 56 above, $\mathcal{O}(1)$ may depend on $m$, but is uniformly bounded away from 0 and $\infty$; more specifically,

$$ 0 < C_{inf}^3 \le \mathcal{O}(1) \le C_{sup}^3 < \infty. $$

**Property 3.9 (Staggered Sequence of $(m, \varepsilon_m^m)$-Neighborhoods).**

Given $m \in \mathbb{N}$, there exists an integer $k_m \in \mathbb{N}$ such that, for each integer $k \ge k_m$, the $(m + k, \varepsilon_{m+k}^{m+k})$-neighborhood of the $m^{th}$ prefractal approximation $\Gamma_{W_m}$ (where $\varepsilon_{m+k}^{m+k}$ is the $(m+k)^{th}$ cohomology infinitesimal, as introduced in Definition 3.1, on page 37),

$$ \mathcal{D}(\Gamma_{W_{m+k}}, \varepsilon_{m+k}^{m+k}) = \{M = (x, y) \in \mathbb{R}^2, d(M, \Gamma_{W_{m+k}}) \le \varepsilon_{m+k}^{m+k}\}, \qquad (\mathbb{R} \ 28) $$

is contained in the $(m, \varepsilon_m^m)$-neighborhood of the $m^{th}$ prefractal approximation $\Gamma_{W_m}$,

$$ \mathcal{D}(\Gamma_{W_m}, \varepsilon_m^m) = \{M = (x, y) \in \mathbb{R}^2, d(M, \Gamma_{W_m}) \le \varepsilon_m^m\}; \qquad (\mathbb{R} \ 29) $$

namely,

$$ \mathcal{D}(\Gamma_{W_{m+k}}, \varepsilon_{m+k}^{m+k}) \subset \mathcal{D}(\Gamma_{W_m}, \varepsilon_m^m). \qquad (\mathbb{R} \ 30) $$

*Proof.* This proof is based on the fact that the sequence of sets of vertices $(V_m)_{m \in \mathbb{N}}$ is increasing (see part i. of Property 2.4, on page 14), and that $V^* = \bigcup_{n \in \mathbb{N}} V_n$ is dense in the Weierstrass Curve $\Gamma_W$, along
where, in each of these two sums, $\omega$ ranges through all of the Complex Dimensions of $\Gamma_W^I$ (i.e., the poles of either $\tilde{\zeta}_m^e$ or, equivalently, $\zeta_m^e$).

Recall that

$$ \mathrm{res}(\zeta_m^e, \omega) = (2 - \omega) \mathrm{res}(\tilde{\zeta}_m^e, \omega). \qquad (56) $$

In order to obtain the fractal tube formula in Theorem 4.1, on page 48, however, we did not need to appeal to the aforementioned results of the general theory, by first calculating $\tilde{\zeta}_{m,\Gamma_m}^e$ or $\zeta_{m,\Gamma_m}^e$ (using their basic scaling and symmetry properties described in [39], along with the geometric properties of $\Gamma_W$ described in Section 2 above) and then, verifying that the appropriate notion of strong languidity is satisfied. This could possibly have been done, although it seems to us very unlikely, but was unnecessary in our present situation.

Instead, as was explained earlier, we first directly calculated the tubular volume $V_m$ in Theorem 4.1, on page 48, and then, after having introduced the associated notion of $m^{th}$ effective volume $\tilde{V}_m$, deduced from the resulting fractal tube formula, via Mellin transformation, an explicit expression for the $m^{th}$ effective prefractal tube zeta function $\tilde{\zeta}_m^e$. Finally, we deduced (in Theorem 4.5, on page 56) the values of the Complex Dimensions of the Weierstrass IFD $\Gamma_W^I$, as the poles of $\tilde{\zeta}_m^e$ (or, equivalently, of $\zeta_m^e$, since $D_W < 2$), for all $m \in \mathbb{N}^*$ sufficiently large.

### 4.3. The noninteger case

An interesting question is the generalization of our previous results to the noninteger case; i.e., to the case when the Weierstrass function $W$ is defined, for any real number $x$, by

$$ W(x) = \sum_{n=0}^{\infty} \lambda^n \cos(2\pi b^n x), \qquad (57) $$

where the real number $b$ does not belong to the set of natural integers.

We plan to provide the details in a later work, but for now limit ourselves to a few comments.

From the geometric point of view, one cannot handle things in the same way. For instance, one cannot resort to a finite IFS, and the function, apart from its parity, has no periodicity property.

Yet, the associated graph being the attractor of the infinite set of maps, $\mathcal{T}_W = \{T_i\}_{i \in \mathbb{Z}}$, such that, for any integer $i$ and $(x, y)$ in $\mathbb{R}^2$,

$$ T_i(x, y) = \left( \frac{x + i}{b}, \lambda y + \cos \left( 2\pi \left( \frac{x + i}{b} \right) \right) \right), \qquad (58) $$

it is natural to consider the associated infinite IFS (IIFS), $\mathcal{T}_W$. As a consequence, the resulting prefractal graphs are infinite ones.
with the fact that the prefractal graph sequence $(\Gamma_{W_m})_{m \in \mathbb{N}}$ converges to the Weierstrass Curve $\Gamma_W$ (for example, in the sense of the Hausdorff metric on $\mathbb{R}^2$).

Given $m \in \mathbb{N}$, there exists an integer $k_{0,m} \in \mathbb{N}$ such that, for each integer $k \ge k_{0,m}$, we have that

$$d(\Gamma_{W_m}, \Gamma_{W_{m+k}}) = \inf_{\substack{0 \le j \le \#V_m - 1 \\ 0 \le j' \le \#V_{m+k} - 1}} \{d(M_{j,m}, M_{j',m+k}), M_{j,m} \in V_m, M_{j',m+k} \in V_{m+k} \setminus V_m\} \le \varepsilon_m^m.$$

We then deduce that for all $k \ge k_{0,m}$,

$$\Gamma_{W_{m+k}} \subset \mathcal{D}(\Gamma_{W_m}, \varepsilon_m^m).$$

At the same time, since, for any $(m, k) \in \mathbb{N}^2$,

$$\varepsilon_{m+k}^{m+k} \le \varepsilon_m^m,$$

along with the fact that, for any $m \in \mathbb{N}$,

$$\lim_{k \to \infty} \varepsilon_{m+k}^{m+k} = 0,$$

we can find another integer $k_{1,m} \in \mathbb{N}$ such that, for each integer $k \ge k_{1,m}$, we have that

$$\mathcal{D}(\Gamma_{W_{m+k}}, \varepsilon_{m+k}^{m+k}) \subset \mathcal{D}(\Gamma_{W_m}, \varepsilon_m^m).$$

The desired result is obtained by letting $k_m = \max\{k_{0,m}, k_{1,m}\}$.

□

**Remark 3.3 (Connection Between Fractality and the Cohomology Infinitesimal).**

As is mentioned in [DL22c], 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. Note that this is in perfect agreement with what is evoked by the French physicist Laurent Nottale in [Not98] 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 $(\Gamma_{W_m})_{m \in \mathbb{N}}$, it is natural – and consistent with the result of Property 3.9, on page 59 above – 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 $(\mathcal{D}(\Gamma_{W_m}, \varepsilon_m^m))_{m \in \mathbb{N}}$ of $\varepsilon_m^m$-neighborhoods of $\Gamma_{W_m}$, where, for each integer $m \in \mathbb{N}$, $\varepsilon_m^m$ is the $m^{th}$ cohomology infinitesimal introduced in Definition 3.1, on page 37 above.
The local Hölder and reverse-Hölder continuity properties of the Weierstrass function then enable us to resort to estimates that are equivalent to the ones obtained in Corollary 2.12, on page 21, and Corollary 2.13, on page 22, and, consequently, to the resulting ones about the elementary heights obtained in Corollary 2.16, on page 25.

As for the effective tubular neighborhood, due to the polygonal approximations in-
duced by the prefractals, it is still obtained by means of rectangles and wedges.

In the integer case, extra terms coming from overlapping rectangles vanished, thanks
to the symmetry with respect to the vertical line $x = \frac{1}{2}$, as described in Lemma 3.9, on
page 45. In the non-integer case, one simply replaces this symmetry with the one with
respect to the vertical axis $x = 0$.

In this light, it is expected that a similar method, suitably adapted, would lead to a
fractal tube formula of the same type as the one obtained in Theorem 4.1, on page 48,
which would then yield the same results concerning the possible Complex Dimensions of
the Weierstrass Curve.

As in the integer case, the terms involving $\varepsilon^{2-D_W+k}(2-D_W)-i\ell\mathbf{p}$ come from the con-
tribution of the rectangles. Hence, much as in Theorem 4.5, on page 56, for all $k \in \mathbb{N}$
and $\ell \in \mathbb{Z}$, the complex numbers $D_W - k(2-D_W) + i\ell\mathbf{p}$, are all simple Complex Di-
mensions of the Weierstrass Curve; i.e., they are simple poles of the effective global tube
zeta function.

We also mention that we could deal with the case $\lambda b < 1$, exactly in the same manner,
and with similar conclusions – although, for example, the abscissae of convergence in-
volved would now correspond to convergence in a left (rather than in a right) half-plane.
Actually, it is noteworthy that, in the present paper, all of our results remain valid (once
suitably adapted) when $\lambda N_b < 1$, where $b = N_b$ is an integer greater than or equal to
two. Observe that in the latter case, the Weierstrass Curve $\Gamma_W$ is of class $C^1$ (or better),
but is still fractal, because it has nonreal Complex Dimensions (in fact, infinitely many
of them).

**5. Concluding comments**

In the light of our results, the box dimension $D_W$ stands as a simple pole of the
effective prefractal tube zeta functions associated to the Weierstrass IFD. It is also
the abscissa of holomorphic continuation of those functions, which therefore cannot be
extended holomorphically to the left of $D_W$; see Theorem 4.2, on page 52, along with
the comments surrounding it.

A natural question which arises is whether the Complex Dimensions of the considered
fractal – in our case, the Weierstrass Curve – are the same as those of the whole set of
prefractal approximations. In [14], by means of the exact sequence of the effective local
fractal zeta functions associated with the sequence of polygonal neighborhoods which
converge to the Curve, we prove that the limit (or global) fractal zeta function has the
same poles as the set of fractal zeta functions associated with the prefractal approxima-
tions. As is shown in [14], the determination of the explicit Complex Dimensions of the
# 4 Prefractal Tube Formulas, Complex Dimensions and Average Minkowski Content

## 4.1 Preliminaries

**Property 4.1 (Fourier Series Expansion of the one-periodic Map $x \mapsto N_b^{-\{x\}}$ [LvF06] ).**

The fractional part map $\{\cdot\}$ is one-periodic. Hence, it is also the case of the map $x \mapsto N_b^{-\{x\}}$, which admits, with respect to the real variable $x$, the following Fourier Series expansion:

$$N_b^{-\{x\}} = \frac{N_b - 1}{N_b} \sum_{\ell \in \mathbb{Z}} \frac{e^{2i\pi m x}}{\ln N_b + 2i\ell\pi} = \frac{N_b - 1}{N_b} \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i\ell \mathbf{p}} (\varepsilon_m^m)^{-i\ell \mathbf{p}}}{\ln N_b + 2i\ell\pi},$$

where, for each $\ell \in \mathbb{Z}$, the exponential Fourier coefficients $c_\ell$ have been obtained through

$$c_\ell = \int_0^1 N_b^{-t} e^{-2i\pi \ell t} dt = \int_0^1 e^{-t \ln N_b} e^{-2i\pi \ell t} dt = -\frac{1}{\ln N_b + 2i\ell\pi} [e^{-t \ln N_b} e^{-2i\pi \ell t}]_{t=0}^1 \\ = \frac{1}{\ln N_b + 2i\ell\pi} [1 - \frac{1}{N_b}] = \frac{N_b - 1}{N_b} \frac{1}{\ln N_b + 2i\ell\pi}.$$

Thus, for any $x \in \mathbb{R}$,

$$N_b^{-\{x\}} = \frac{N_b - 1}{N_b} \sum_{\ell \in \mathbb{Z}} \frac{e^{2i\pi m x}}{\ln N_b + 2i\ell\pi}.$$

Note that since

$$x = -\ln N_b ((N_b - 1) \varepsilon_m^m),$$

one has, for every $m \in \mathbb{Z}$,

$$e^{2i\pi m x} = e^{-2i\pi m \ln N_b ((N_b - 1) \varepsilon_m^m)} = e^{-2i\pi m \frac{\ln((N_b - 1)\varepsilon_m^m)}{\ln N_b}}.$$

## Definition 4.1 (Oscillatory Period).

Following [LvF00], [LvF06], [LRŽ17b], we introduce the oscillatory period of the Weierstrass IFD:

$$\mathbf{p} = \frac{2\pi}{\ln N_b}.$$

## Definition 4.2 ($\ell^{th}$-order Vibration Mode).

Given $\ell \in \mathbb{Z}$, we define the $\ell^{th}$-order vibration mode as the one associated to $\ell \mathbf{p}$.
IFD is a compulsory step in order to obtain the Complex Dimensions of the limit fractal Curve. This fact provided a key step in the proof of Theorem 4.3, on page 54, and of Theorem 4.5, on page 56.

To this point, one may question the usefulness of the tubular approach in such a complicated geometry, where the polygonal approach seems to provide direct results. As a matter of fact, the tubular approach is required in the subcritical case of smooth fractals (at least $C^1$-smooth), which happens, for instance, when $\lambda N_b < 1$ (see our recent work [10]). In that case, a polygonal approach is not useful. Note that such problems, for the determination of the Complex Dimensions, are in direct connection with real life. For instance, the plasma membrane of the slime mold Physarum polycephalum presents an irregular appearance with numerous invaginations at multiple scales. This membrane is a dynamic structure, constantly changing as the organism moves, feeds itself, and explores its environment. It displays a complex, adaptive behavior that might be described as fractal-like evidence, for which the determination of the associated Complex-Dimension is crucial in terms of the understanding of its growth process.

Now, as was alluded to in the Introduction, the determination of the possible Complex Dimensions of a fractal object, being deeply connected with its intrinsic vibrational properties, is thus directly associated to its cohomological properties: what are the topological invariants of the Weierstrass Curve? This is the question we try to answer in the second part of our study, [12], where we determine the fractal cohomology of the Weierstrass Curve. Remarkably the results obtained in [13] and in [12] are fully compatible with those obtained in the present paper.

Behind the fractal series expansion of the Weierstrass function lies another expansion, indexed by the Complex Dimensions obtained in our fractal tube formulas (see Theorem 4.1, on page 48). Intuitively, one understands that the terms of the expansion come from the cohomology groups associated to the prefractal sequence of finite graphs that converges towards the Curve. This is all the more interesting, as those groups possess the same symmetries as the Curve, which means that a specific complex differentiation could be achieved on this, however, everywhere singular object; see [9], [10] and [12].

As was mentioned in Section 4.3, on page 59, we also intend, in a future work, to extend our results to the general case, i.e., when the Weierstrass function $W$ is defined, for any real number $x$, by

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

where the real number $b$ does not belong to the set of natural integers. This goes along with a generalization of the results of the present paper to a large class of Weierstrass-like functions (see the paper [5]), including the Takagi function, the Knopp functions and the Koch parametrized Curve; see [11], [8].
**Property 4.2.** For any integer $\ell \in \mathbb{Z}$, as given in Property 4.1, on page 61,

$$e^{2i\pi\ell x} = e^{-i\ell\mathbf{p}\ln((N_b-1)\varepsilon_m^m)} = ((N_b-1)\varepsilon_m^m)^{-i\ell\mathbf{p}} = (N_b-1)^{-i\ell\mathbf{p}}(\varepsilon_m^m)^{-i\ell\mathbf{p}},$$

where the oscillatory period $\mathbf{p}$ has been introduced in Definition 4.1, on page 61 above.

Thus, for any $x \in \mathbb{R}$ and any $\varepsilon_m^m > 0$,

$$N_b^{-\{x\}} = \frac{N_b - 1}{N_b} \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i \ell \mathbf{p}} (\varepsilon_m^m)^{-i \ell \mathbf{p}}}{\ln N_b + 2i \ell \pi} , \quad N_b^{-D_W \{x\}} = \frac{N_b^{D_W} - 1}{N_b^{D_W}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{D_W} - 1)^{-i \ell \mathbf{p}} (\varepsilon_m^m)^{-i \ell \mathbf{p}}}{D_W \ln N_b + 2i \ell \pi}.$$

**Definition 4.3 (Effective Distance and Tube Zeta Functions Associated to an Arbitrary IFD of $\mathbb{R}^2$).**

Let $\mathcal{F}^I$ be an iterated fractal drum of $\mathbb{R}^2$; i.e., given a cohomology infinitesimal $\varepsilon_{\mathcal{F}} = (\varepsilon_{m,\mathcal{F}}^m)_{m \in \mathbb{N}}$, as introduced in Definition 3.2, on page 38, $\mathcal{F}^I$ is a 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 approximation to a fractal curve $\mathcal{F}$.

We are assuming here that $(\varepsilon_{m,\mathcal{F}}^m)_{m \in \mathbb{N}}$ is a decreasing sequence of positive numbers tending to 0 as $m \to \infty$, such that, for all fixed $m \in \mathbb{N}$, $\lim_{k_m \to \infty} (\varepsilon_{m,\mathcal{F}}^m - \varepsilon_{m+k_m,\mathcal{F}}^{m+k_m}) = 0$. This is the case, in particular, for the Weierstrass IFD, according to Definition 3.1, on page 37. Indeed, with the notation of the latter definition, we have that $\varepsilon_{m+1,\mathcal{F}}^{m+1} = \varepsilon_{m+1}^{m+1} = \frac{\varepsilon_m^m}{N_b}$, for all $m \in \mathbb{N}$. Hence, for any $k_m \in \mathbb{N}$, $\varepsilon_{m+k_m,\mathcal{F}}^{m+k_m} = \varepsilon_{m+k_m}^{m+k_m} = \frac{\varepsilon_m^m}{N_b^{k_m}}$.

Back to the general case of $\mathcal{F}^I$, we hereafter consider the $\varepsilon_{m,\mathcal{F}}^m$-neighborhood (or $\varepsilon_{m,\mathcal{F}}^m$-tubular neighborhood) of $\mathcal{F}_m$,

$$\mathcal{D}(\mathcal{F}_m, \varepsilon_{m,\mathcal{F}}^m) = \{M \in \mathbb{R}^2, d(M, \mathcal{F}_m) \le \varepsilon_{m,\mathcal{F}}^m\}, \quad (\mathbb{R} 31)$$

of tubular volume (i.e., area) denoted $\mathcal{V}_{m,\mathcal{F}_m}(\varepsilon_{m,\mathcal{F}}^m)$.

In our present context, when it comes to obtaining the associated fractal tube zeta function, we cannot, as in the case of an arbitrary bounded subset of $\mathbb{R}^2$ (see [LRŽ17b], Definition 2.2.8, page 118), directly use an integral formula of the form

$$\tilde{\zeta}_{m, \mathcal{F}_m}(s) = \int_0^{\varepsilon_{m, \mathcal{F}}^m} t^{s-3} \mathcal{V}_{m, \mathcal{F}_m}(t) dt = \int_0^{\varepsilon_{m, \mathcal{F}}^m} t^{s-2} \mathcal{V}_{m, \mathcal{F}_m}(t) \frac{dt}{t}, \quad (\mathbb{R} 32)$$

since the tube formulas that we will obtain in Subsection 4.2 below can only be expressed in an explicit way at a cohomology infinitesimal.

However, if, instead of considering the tube volume function at a given step $m_0$, we consider the more general tube volume functions for all integers $m \ge m_0$, we can then use Riemann sums, for the following nonuniform partition of the interval $[0, \varepsilon_{m,\mathcal{F}}^m]$: given an arbitrary nonnegative integer $k_m \ge m$,

$$[0, \varepsilon_{m,\mathcal{F}}^m] = [0, \varepsilon_{m k_m, \mathcal{F}}^m] \cup \left\{ \bigcup_{m+k_m+p=m}^{m+k_m+p=m+k_m} \left[ \varepsilon_{m+k_m+p+1, \mathcal{F}}^{m+k_m+p+1}, \varepsilon_{m+k_m+p, \mathcal{F}}^{m+k_m+p} \right] \right\} \cup \left[ \varepsilon_{m+k_m, \mathcal{F}}^{m+k_m}, \varepsilon_{m, \mathcal{F}}^m \right], \quad (\mathbb{R} 33)$$
Acknowledgments

The authors would like to thank Goran Radunović and Steffen Winter for their helpful questions about the first version of this paper. We are also very grateful to the anonymous referee for his very careful reading, comments and suggestions, which significantly helped to improve the quality of the paper.

**Appendix A. Technical appendix**

In this Appendix, subdivided into Appendices A.1–A.6, we provide the proof of several key results needed in order to establish our main results.

A.1. *Proof of Property 2.10, stated on page 20*

**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 $0 \le j \le \frac{N_b - 1}{2}$:

$$0 \le \frac{2\pi j}{N_b - 1} \le \pi, \quad 0 \le \frac{2\pi(j+1)}{N_b - 1} \le \pi \left(1 + \frac{2}{N_b - 1}\right).$$

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 $0 \le \frac{2\pi j}{N_b - 1} \le \frac{2\pi(j+1)}{N_b - 1} \le \pi$.
where the second union is taken over all $p \in \mathbb{N}$ satisfying the indicated relations and equalities.

Note that the mesh of the partition (i.e., the length of the largest subinterval) is

$$ \varepsilon_{m,\mathcal{F}}^{m} - \varepsilon_{m+k,\mathcal{F}}^{m+k_{m}}, \quad \text{with} \quad \lim_{k_{m} \to \infty} (\varepsilon_{m,\mathcal{F}}^{m} - \varepsilon_{m+k_{m},\mathcal{F}}^{m+k_{m}}) = 0. $$

We set

$$ \sigma_{0,m k_m} = \varepsilon_{m k_m, \mathcal{F}}^{m k_m}, \quad \sigma_{m,m+k_m} = \varepsilon_{m, \mathcal{F}}^{m} - \varepsilon_{m+k_m, \mathcal{F}}^{m+k_m}, \qquad (\mathcal{R} 34) $$

and, for all $p \in \{0, \dots, m k_m - m - k_m - 2\}$,

$$ \sigma_p = \varepsilon_{m+k_m+p, \mathcal{F}}^{m+k_m+p} - \varepsilon_{m+k_m+p+1, \mathcal{F}}^{m+k_m+p+1}. \qquad (\mathcal{R} 35) $$

We can then introduce the Riemann sum

$$ \begin{aligned} \operatorname{Riemann}(k_m, m, \mathcal{V}) = & \sigma_0 (\varepsilon_{m m, \mathcal{F}}^{m k_m})^{s-3} \mathcal{V}_{m k_m, \mathcal{F}_{m k_m}} (\varepsilon_{m k, \mathcal{F}}^{m k_m}) + \sigma_{m+k_m} (\varepsilon_{m, \mathcal{F}}^{m})^{s-3} \mathcal{V}_{m k_m, \mathcal{F}_{m k_m}} (\varepsilon_{m, \mathcal{F}}^{m}) \\ & + \sum_{0 \le p \le m} \sigma_p (\varepsilon_{m+k_m+p, \mathcal{F}}^{m+k_m+p})^{s-3} \mathcal{V}_{m+k_m+p+1, \mathcal{F}_{m+k_m+p+1}} (\varepsilon_{m+k_m+p, \mathcal{F}}^{m+k_m+p}). \end{aligned} \qquad (\mathcal{R} 36) $$

We then have that

$$ \lim_{k_m \to \infty} \operatorname{Riemann}(k_m, m, \mathcal{V}_m) = \int_0^{\varepsilon_{m, \mathcal{F}}^m} t^{s-2} \tilde{\mathcal{V}}_{m, \mathcal{F}_m}(t) \frac{dt}{t}, \qquad (\mathcal{R} 37) $$

where, for all sufficiently large $n \in \mathbb{N}^*$, $\tilde{\mathcal{V}}_{n, \mathcal{F}_n}$ is the continuous function defined for all $t \in [0, \varepsilon_{n, \mathcal{F}}^n]$ by substituting $t$ for $\varepsilon_{n, \mathcal{F}}^n$ in the expression for $\mathcal{V}_{n, \mathcal{F}_n} (\varepsilon_{n, \mathcal{F}}^n)$.

For the fractal power series obtained for $\tilde{\mathcal{V}}_{n, \mathcal{F}_n}$ in the case of the Weierstrass IFD of key interest to us in this paper, see Theorem 4.7, on page 72, in Subsection 4.2 below.

One can think of $\tilde{\mathcal{V}}_{n, \mathcal{F}_n} = \tilde{\mathcal{V}}_{n, \mathcal{F}_n}(t)$ as being the effective volume of the $t$-neighborhood of the $n^{th}$ prefractal approximation to the fractal curve defined by the IFD $\mathcal{F}$, for every value of $t \in [0, \varepsilon_{n, \mathcal{F}}^n]$. Note that, in fact, $\tilde{\mathcal{V}}_{n, \mathcal{F}_n} = \tilde{\mathcal{V}}_{n, \mathcal{F}_n}(t)$ is given by the same (locally uniformly) convergent fractal power series for all nonzero $t \in \mathbb{C}$ with $|t| \le \varepsilon_{n, \mathcal{F}}^n$ (by convention, $\tilde{\mathcal{V}}_{n, \mathcal{F}_n}(0) = \mathcal{V}_{n, \mathcal{F}_n}(0) = 0$); however, we will not make use of this observation in this paper.

We can then obtain the resulting $m^{th}$ effective local tube zeta function $\tilde{\zeta}_{m, \mathcal{F}}^e$ – a generalization to IFDs of the usual definition referred to just above – defined for all $s$ in $\mathbb{C}$ with sufficiently large real part (in fact, for $\mathcal{R}e(s) > D_{m, \mathcal{F}_m}$, where $D_{m, \mathcal{F}_m}$ is the abscissa of convergence of $\tilde{\zeta}_{m, \mathcal{F}_m}$), by

$$ \tilde{\zeta}_{m, \mathcal{F}_m}^e(s) = \lim_{k \to \infty} \int_0^{\varepsilon_{m, \mathcal{F}}^m} t^{s-3} \tilde{\mathcal{V}}_{m, \mathcal{F}_m}(t) dt = \int_0^{\varepsilon_{m, \mathcal{F}}^m} t^{s-2} \tilde{\mathcal{V}}_{m, \mathcal{F}_m}(t) \frac{dt}{t}. \qquad (\mathcal{R} 38) $$

Note that still for $\mathcal{R}e(s) > D_{m, \mathcal{F}}$, $\tilde{\zeta}_{m, \mathcal{F}_m}$ is also given by a limit of Riemann sums, as given in the identity (R36), on page 63. This statement follows by combining relations (R38) and (R37)

As for the $m^{th}$ effective local distance zeta function $\zeta_{m, \mathcal{F}}$, it can be deduced by the following functional equation (in the present case, when $\mathcal{F}_m \subset \mathbb{R}^2$), for the same value of $\varepsilon_{m, \mathcal{F}}^m > 0$,

$$ \zeta_{m, \mathcal{F}_m}^e(s) = (\varepsilon_{m, \mathcal{F}}^m)^{s-2} \mathcal{V}_{m, \mathcal{F}_m} (\varepsilon_{m, \mathcal{F}}^m) + (2-s) \tilde{\zeta}_{m, \mathcal{F}_m}^e(s) \cdot \quad (\blacklozenge) \qquad (\mathcal{R} 39) $$
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:

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

### A.2. Proof of Property 2.11, stated on page 21

**Proof.** i. One simply applies Proposition 2.3, on page 11, in conjunction with Property 2.9, on page 19.

For $i$ in $\{0, \dots, N_b - 1\}$, we have that

$$\begin{align*}
& T_i \left( x_{j-i} (N_b-1) N_b^{m-1}, m-1 \right), W \left( x_{j-i} (N_b-1) N_b^{m-1}, m-1 \right) \\
& \qquad \| \\
& \left( x_{j-i} (N_b-1) N_b^{m-1}, m + \frac{i}{N_b}, \lambda W \left( x_{j-i} (N_b-1) N_b^{m-1}, m-1 \right) \right. \\
& \qquad \left. + \cos \left( 2\pi \left( x_{j-i} (N_b-1) N_b^{m-1}, m + \frac{i}{N_b} \right) \right) \right) \\
& = (x_{j,m}, W(x_{j,m-1} - i) + \cos(2\pi x_{j,m})) \\
& = \left( x_{j,m}, W(x_{j,-1} - i) + \cos \left( 2\pi x_{j-i,m} + \frac{i}{N_b} \right) \right) \\
& = (x_{j,m}, \lambda W(x_{j,m-1}) + \cos(2\pi x_{j-i,m})) \\
& = (x_{j,m}, W(x_{j,m})).
\end{align*}$$
Observe that since, by construction,

$$V_{m, \mathcal{F}_m} (\varepsilon_{m, \mathcal{F}}^m) = \tilde{V}_{m, \mathcal{F}_m} (\varepsilon_{m, \mathcal{F}}^m),$$

the associated functional equation of relation (R39) just above is the exact analog of the functional equation connecting the usual tube and zeta functions of a bounded set (or, more generally, of a related fractal drum) in the standard higher-dimensional of Complex Dimensions developed in [LRŽ17b], as well as in [LRŽ17a], [LRŽ17c] and [LRŽ18].

Remark 4.1. We stress the fact that $\tilde{\zeta}_{m, \mathcal{F}_m}^e$ does not coincide with the usual tube zeta function $\tilde{\zeta}_{\mathcal{F}_m}$ associated with the $m^{th}$ polygonal prefractal approximation $\mathcal{F}_m \subset \mathbb{R}^2$ to the fractal curve $\mathcal{F}$, given, as in [LRŽ17b], for all $s \in \mathbb{C}$ with $\mathcal{R}e(s)$ sufficiently large, by

$$\tilde{\zeta}_{\mathcal{F}_m}(s) = \int_0^{\varepsilon_{m, \mathcal{F}}^m} t^{s-3} V_{m, \mathcal{F}_m}(t) dt = \int_0^{\varepsilon_{m, \mathcal{F}}^m} t^{s-2} V_{m, \mathcal{F}_m}(t) \frac{dt}{t}.$$

Similarly, $\zeta_{m, \mathcal{F}_m}^e$ does not coincide with the usual distance zeta function $\zeta_{\mathcal{F}_m}$ associated with the $m^{th}$ polygonal prefractal approximation $\mathcal{F}_m \subset \mathbb{R}^2$ to the fractal curve $\mathcal{F}$, given, as in [LRŽ17b], for all $s \in \mathbb{C}$ with $\mathcal{R}e(s)$ sufficiently large, by

$$\zeta_{\mathcal{F}_m}(s) = \int_{M \in \mathcal{D}(\Gamma_{\mathcal{F}_m}, \varepsilon_{m, \mathcal{F}}^m)} (d(M, \mathcal{F}_m))^{s-2} dt,$$

where $\mathcal{D}(\Gamma_{\mathcal{F}_m}, \varepsilon_{m, \mathcal{F}}^m)$ is the $\varepsilon_{m, \mathcal{F}}^m$-neighborhood (or $\varepsilon_{m, \mathcal{F}}^m$-tubular neighborhood) of $\mathcal{F}_m$, given by

$$\mathcal{D}(\Gamma_{\mathcal{F}_m}, \varepsilon_{m, \mathcal{F}}^m) = \{M \in \mathbb{R}^2, d(M, \mathcal{F}_m) \leq \varepsilon_{m, \mathcal{F}}^m\}.$$

This entire comment applies, in particular, to the Weierstrass IFD, which is the central object of this paper.

**Notation 11 (Natural Volume Extension).**

For the sake of simplicity, given $m \in \mathbb{N}$, we will from now on call the $m^{th}$ natural volume extension, the volume extension function $\tilde{V}_{m, \mathcal{F}_m}$ associated with $V_{m, \mathcal{F}_m}$. Alternatively, as was mentioned earlier, $\tilde{V}_{m, \mathcal{F}_m}$ will be called the $m^{th}$ effective tubular volume. This notation and terminology apply, in particular, to the different volume functions involved in the discussion of the Weierstrass IFD in Subsection 4.2 below.

**Remark 4.2 (Consistency of our Approach in the Case of the Weierstrass IFD – Connection with Reality).**

As shown in Remark 3.1, on page 37, the $m^{th}$ prefractal approximations to the Weierstrass Curve become closer and closer as $m$ increases. Hence, it makes sense to consider a continuous version of the tubular volume, where the discrete and the continuous, in a sense, eventually merge, for all $m \in \mathbb{N}^*$ sufficiently large.

We can also note that, in real life, fractality is not always the result of a discrete process. On the contrary, fractal shapes develop continuously, as is the case, for instance, in biology.
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 $\{0, \dots, N_b - 1\}$:

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

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

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

Henceforth, we want to prove that, for any integer $k$ in $\{0, \dots, N_b^{m-1} - 1\}$,

$$ \operatorname{sgn}\left(\mathcal{W}\left(x_{k\left(N_b-1\right)+j+1,m}\right) - \mathcal{W}\left(x_{k\left(N_b-1\right),j,m}\right)\right) = \operatorname{sgn}\left(\mathcal{W}\left(\frac{j+1}{N_b-1}\right) - \mathcal{W}\left(\frac{j}{N_b-1}\right)\right). $$
Remark 4.3. It follows from the above relation (R39), on page 63, along with the results (and their proofs) in [LRZ17b], Corollary 2.2.20, on page 127, that, in a given domain of C, the effective fractal zeta functions $\zeta_{m,\mathcal{F}_m}^e$ and $\tilde{\zeta}_{m,\mathcal{F}_m}^e$ have the same poles (denoted by $\omega$) with residues connected by the relation

$$ \mathrm{res} \left( \tilde{\zeta}_{m, \mathcal{F}_m}^e, \omega \right) = \frac{1}{2-\omega} \mathrm{res} \left( \zeta_{m, \mathcal{F}_m}^e, \omega \right), \quad (\blacklozenge \blacklozenge) \qquad (\mathcal{R} 40) $$

in case $\omega \neq 2$ is a simple pole; and, similarly for the principal parts of $\zeta_{m,\mathcal{F}_m}^e$ and $\tilde{\zeta}_{m,\mathcal{F}_m}^e$ at $\omega$, in case $\omega \neq 2$ is a multiple pole. It follows, in particular, that, in the present new sense, the Complex Dimensions of $\mathcal{F}_m$ can be indifferently defined as the (visible) poles of the effective distance zeta function $\zeta_{m,\mathcal{F}_m}^e$ or of the effective tube zeta function $\tilde{\zeta}_{m,\mathcal{F}_m}^e$.

We will show in Subsection 4.2 below that, in the case of the Weierstrass IFD, and for all integers $m$ sufficiently large, $\tilde{\zeta}_{m,\mathcal{F}_m}^e$ (and hence also, $\zeta_{m,\mathcal{F}_m}^e$, in light of relation (R40), on page 65 above), has a meromorphic continuation to all of $C$ and has Minkowski dimension strictly smaller than 2; so that its Complex Dimensions are simple and have real part strictly smaller than 2. Hence, for any Complex Dimension $\omega$ of the Weierstrass IFD, we have that $\omega$ is simple and $\omega \neq 2$. (See, especially, Theorem 4.8, on page 76, and Theorem 4.10, on page 81, along with Corollary 4.9, on page 80).

## 4.2 Prefractal Tube Formulas and Prefractal Effective Zeta Functions

In order to obtain the main results of this section – namely, Theorem 4.7, on page 72, Theorem 4.8, on page 76, and 4.11, on page 83, along with Corollary 4.9, on page 80, and Theorem 4.10, on page 81 below, 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 $\Gamma_{W_m}$, here, the rectangles and the wedges (in Property 4.3, on page 65, and Property 4.4, on page 68 respectively), thereby supplementing the study of the positive or negative contributions of the rectangles, triangles and extreme wedges carried out earlier in Section 3, and synthetized in Propositions 3.5–3.8, on pages 56–59 above. 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 subsection.

In the sequel, in the case when $\mathcal{F}$ is the Weierstrass IFD, we will write, for example, $\tilde{\mathcal{V}}_{m,\Gamma_{W_m}}$, $\mathcal{V}_{m,\Gamma_{W_m}}$, $\tilde{\zeta}_{m,\Gamma_W}^e$, $\zeta_{m,\Gamma_W}^e$, instead of $\tilde{\mathcal{V}}_{m,\mathcal{F}_m}$, $\mathcal{V}_{m,\mathcal{F}_m}$, $\tilde{\zeta}_{m,\mathcal{F}_m}^e$, $\zeta_{m,\mathcal{F}_m}^e$, respectively. And similarly for the corresponding expressions associated with the contributions of the rectangles, wedges, outer triangles and parallelograms, for instance, as in Section 3 above.

### Property 4.3 (Tube Formula and Effective Tube Zeta Function Associated to the Contribution of the Rectangles to the Tubular Volume).

Given $m \in \mathbb{N}^*$, with Notation 11, on page 64, the contribution $\tilde{\mathcal{V}}_{m,\Gamma_{W_m},\mathrm{Rectangles}}(\varepsilon_m^m)$ of the $2(N_b - 1)N_b^m$ rectangles to the effective tubular volume $\tilde{\mathcal{V}}_{m,\Gamma_{W_m}}(\varepsilon_m^m)$ is

$$ \begin{aligned} \tilde{\mathcal{V}}_{m,\Gamma_{W_m},\mathrm{Rectangles}}(\varepsilon_m^m) &= 2 \sum_{k=0}^{\infty} \binom{\frac{1}{2}}{k} N_b^{-(1-k(2-D_W))\{x\}} (\varepsilon_m^m)^{2-D_W+k(2-D_W)} \mathcal{O}(1) \\ &= 2 \sum_{k=0}^{\infty} \binom{\frac{1}{2}}{k} \frac{N_b^{1-k(2-D_W)} - 1}{N_b^{1-k(2-D_W)}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i\ell\mathbf{p}} (\varepsilon_m^m)^{2-D_W+k(2-D_W)-i\ell\mathbf{p}}}{(1-k(2-D_W))\ln N_b + 2i\ell\pi} \mathcal{O}(1). \end{aligned} \qquad (\mathcal{R} 41) $$
The induction hypothesis will be used in so far as any $k$ in $\{0, \dots, N_b^m - 1\}$ 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

$$W(x + i) = W(x).$$

Due to the symmetry with respect to the vertical line $x = \frac{1}{2}$ (see Property 2.1, on page 10), given a nonnegative 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

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

Thus, we only have to consider the cases when

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

The remaining ones, namely, the cases when

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

are then obtained by symmetry.

Hence,

$$W(x_k(N_b-1)+j+1,m) - W(x_{j,m})$$
Recall from the discussion in Subsection 4.1 that, by construction,

$$ \tilde{V}_{m, \Gamma_{W_m}, \text{Rectangles}} (\varepsilon_m^m) = V_{m, \Gamma_{W_m}, \text{Rectangles}} (\varepsilon_m^m) . $$

For the sake of clarity, and in order to avoid confusion between various occurrences of $O(1)$, we will write relation (R41) in the form

$$ \tilde{V}_{m, \Gamma_{W_m}, \text{Rectangles}} (\varepsilon_m^m) = C_{\text{Rectangles}} \sum_{k=0}^{\infty} \binom{\frac{1}{2}}{k} \frac{N_b^{1-k(2-D_W)} - 1}{N_b^{1-k(2-D_W)}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i\ell \mathbf{p}} (\varepsilon_m^m)^{2-D_W+k(2-D_W)-i\ell \mathbf{p}}}{(1-k(2-D_W)) \ln N_b + 2i\ell\pi} , \quad (R42) $$

where $C_{\text{Rectangles}}$ denotes a strictly positive and finite constant, depending on $m \in \mathbb{N}^*$, but uniformly bounded away from 0 and $\infty$ (i.e., here and in the sequel, independently of $m \in \mathbb{N}^*$ large enough); see Proposition 3.5, on page 56.

The associated $m^{th}$ (local) effective tube zeta function (see Definition 4.3, on page 62 above) is first obtained, for any complex number $s$ such that $\text{Re}(s) > D_W$, as follows:

$$ \begin{aligned} \tilde{\zeta}_{m, \text{Rectangles}}^{e}(s) &= \int_{0}^{\varepsilon_{m}^{m}} t^{s-3} \tilde{V}_{m, \Gamma_{W_m}, \text{Rectangles}}(t) dt \\ &= C_{\text{Rectangles}} \sum_{k=0}^{\infty} \binom{\frac{1}{2}}{k} \frac{N_b^{1-k(2-D_W)} - 1}{N_b^{1-k(2-D_W)}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i\ell \mathbf{p}}}{(1-k(2-D_W)) \ln N_b + 2i\ell\pi} \int_{0}^{\varepsilon_{m}^{m}} t^{s-3} t^{2-D_W+k(2-D_W)-i\ell \mathbf{p}} dt \\ &= C_{\text{Rectangles}} \sum_{k=0}^{\infty} \binom{\frac{1}{2}}{k} \frac{N_b^{1-k(2-D_W)} - 1}{N_b^{1-k(2-D_W)}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i\ell \mathbf{p}}}{(1-k(2-D_W)) \ln N_b + 2i\ell\pi} \frac{(\varepsilon_m^m)^{s-D_W+k(2-D_W)-i\ell \mathbf{p}}}{s - D_W + k(2 - D_W) - i \ell \mathbf{p}} . \end{aligned} \quad (R43) $$

where, as in Notation 11, on page 64 (and the discussion preceding it), $\tilde{V}_{m, \Gamma_{W_m}, \text{Rectangles}}$ is the natural volume extension function associated with $V_{m, \Gamma_{W_m}, \text{Rectangles}}$, which means that $\tilde{V}_{m, \Gamma_{W_m}, \text{Rectangles}}$ is the function defined for all $t \in [0, \varepsilon_m^m]$ by substituting $t$ for $\varepsilon_m^m$ in the fractal power series obtained for $V_{m, \text{Rectangles}}(\varepsilon_m^m)$ in relation (R42), or, equivalently, in relation (R41).

We call this zeta function $\tilde{\zeta}_{m, \text{Rectangles}}^e$ the $m^{th}$ local effective tube zeta function (associated with the rectangles), because it is the zeta function associated **not only** with the $m^{th}$ prefractal approximation to the Weierstrass Curve $\Gamma_W$, but, also, with the infinitesimal $\varepsilon_m^m$ which conveys the scaling relation associated to the limit fractal object; i.e., $\Gamma_W$. The same comment holds for the forthcoming local zeta functions introduced in Properties 4.4–4.6, on pages 68–71.

By meromorphic continuation to all of $\mathbb{C}$, one then obtains the (local) effective tube zeta function $\tilde{\zeta}_{m, \text{Rectangles}}^e$ for all $s \in \mathbb{C}$, as given by the last two equalities in relation (R43) just above.

Furthermore, the abscissa of absolute convergence of the Dirichlet-type integral (DTI) involved in the definition of $\tilde{\zeta}_{m, \text{Rectangles}}^e$, in the sense of $[LR\tilde{Z}17b]$ (Appendix A), is equal to $D_W$.

The associated Complex Dimensions arise as

$$ D_W - k(2 - D_W) + i\ell \mathbf{p} , \quad \text{with } k \in \mathbb{N}, \ell \in \mathbb{Z} . $$
$$ = \lambda \left( W \left( x_k \left(N_b-1\right)+j+1, m-1 \right) - W \left( x_k \left(N_b-1\right)+j, m-1 \right) \right) + \cos \left( 2\pi x_k \left(N_b-1\right)+j+1, m-1 \right) - \cos \left( 2\pi x_k \left(N_b-1\right)+j, m-1 \right) $$

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

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

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

$$ = \lambda \left( W \left( i + x_{\tilde{k}} \left(N_b-1\right)+j+1, m-1 \right) - W \left( i + x_{\tilde{k}} \left(N_b-1\right)+j, m-1 \right) \right) $$

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

$$ = \lambda \left( W \left( x_{\tilde{k}} \left(N_b-1\right)+j+1, m-1 \right) - W \left( x_{\tilde{k}} \left(N_b-1\right)+j, m-1 \right) \right) $$

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

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

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

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,
Note that, thanks to the results of Corollary 2.15, on page 27, the one-periodic function (with respect to the variable $\ln_{N_b}(\varepsilon_m^m)^{-1}$, see Property 4.1, on page 61), associated to the above Complex Dimensions with real part $D_W$, is bounded between two strictly positive and finite constants.

**Remark 4.4.** In the proof of Theorem 4.8, on page 76, we will show that the series appearing on the right-hand side of the expression of $\tilde{V}_{m,\Gamma_{W_m},\text{Rectangles}}(\varepsilon_m^m)$ in formulas (R41)-(R42) in Property 4.3, on page 65 (for all $m \ge 1$ large enough) is absolutely convergent – and hence also, convergent. We will also explain how to derive the expression for the tube zeta function $\tilde{\zeta}_{m,\text{Rectangles}}$ (again, for all $m \ge 1$ large enough), via an application of the (truncated) Mellin transform to the function $t \mapsto \tilde{V}_{m,\text{Rectangles}}(t)$, defined for all $t \in [0, \varepsilon_m^m]$, followed by meromorphic continuation to all of $\mathbb{C}$. We refer to that same proof for the other statements concerning $\tilde{\zeta}_{m,\text{Rectangles}}^e$ and the associated (possible) poles (i.e., the Complex Dimensions of the Weierstrass IFD).

An entirely similar comment could be made (still for all $m \ge 1$ sufficiently large) about $\tilde{V}_{m,\Gamma_{W_m},\text{wedges}}(\varepsilon_m^m)$ and $\tilde{\zeta}_{m,\text{wedges}}^e$ in Property 4.4, on page 68, $\tilde{V}_{m,\Gamma_{W_m},\text{extra outer triangles}}(\varepsilon_m^m)$ and $\tilde{\zeta}_{m,\text{extra outer triangles}}^e$ in Property 4.5, on page 70, $\tilde{V}_{m,\Gamma_{W_m},\text{parallelograms}}(\varepsilon_m^m)$ and $\tilde{\zeta}_{m,\text{parallelograms}}^e$ in Property 4.6, on page 71, as well as about

$$ \tilde{V}_{m,\Gamma_{W_m}}(\varepsilon_m^m) = \tilde{V}_{m,\Gamma_{W_m},\text{Rectangles}}(\varepsilon_m^m) + \tilde{V}_{m,\Gamma_{W_m},\text{wedges}}(\varepsilon_m^m) + \tilde{V}_{m,\Gamma_{W_m},\text{extra outer triangles}}(\varepsilon_m^m) + \tilde{V}_{m,\Gamma_{W_m},\text{parallelograms}}(\varepsilon_m^m), \quad (\mathcal{R} 44) $$

and

$$ \tilde{\zeta}_{m,W_m}^e(s) = \tilde{\zeta}_{m,\text{Rectangles}}^e(s) + \tilde{\zeta}_{m,\text{wedges}}^e(s) + \tilde{\zeta}_{m,\text{extra outer triangles}}^e(s) + \tilde{\zeta}_{m,\text{parallelograms}}^e(s), \quad (\mathcal{R} 45) $$

in Theorem 4.7, on page 72, and Theorem 4.8, on page 76.

**Remark 4.5.** Recall from [LRŽ17b] that the abscissa of convergence $\sigma_m$ of $\tilde{\zeta}_{m,\text{Rectangles}}^e$ is the unique (possibly extended) real number $\sigma_m$ such that the DTI defining $\tilde{\zeta}_{m,\text{Rectangles}}^e$ (in the first equality in relation (R43) above, on page 66), converges for $\mathcal{R}e(s) > \sigma_m$ and diverges for $\mathcal{R}e(s) < \sigma_m$. Here, in the light of the identity (R43), we have that $\sigma_m = D_W$, for all $m \in \mathbb{N}^*$ large enough. An analogous comment applies to all the other DTIs encountered in this subsection, and in Subsection 4.3, including, especially, $\tilde{\zeta}_{m,\text{wedges}}^e$, $\tilde{\zeta}_{m,\text{extra outer triangles}}^e$, $\tilde{\zeta}_{m,\text{parallelograms}}^e$.
$$W(x_{\tilde{k}(N_b-1)+j+1,m-1}) - W(x_{\tilde{k}(N_b-1)+j,m-1}) \le 0.$$

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

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

as required. □

### A.3. Proof of Corollary 2.17, stated on page 26

**Proof.** i. One simply writes, successively:

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

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

ii. One also has

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

where

$$C_{inf} L_m^{2-D_W} \le h_{j-1,m} \le C_{sup} L_m^{2-D_W}.$$

This yields

$$\frac{N_b - 1}{C_{sup}} L_m^{D_W - 1} \le \frac{(N_b - 1) L_m}{h_{j-1,m}} \le \frac{N_b - 1}{C_{inf}} L_m^{D_W - 1}.$$

Consequently, $\theta_{j-1,j,m+1} \lesssim L_m^{D_W - 1}$, as claimed. □
**Property 4.4 (Tube Formula and Effective Tube Zeta Function Associated to the Contribution of the Wedges to the Tubular Volume).**

Given $m \in \mathbb{N}^*$ sufficiently large, with Notation 11, on page 64, the contribution of the wedges to the effective tubular volume $\tilde{\mathcal{V}}_{m,\Gamma_{W_m}}(\varepsilon_m^m)$ is

$$
\begin{align*}
\tilde{\mathcal{V}}_{m,\Gamma_{W_m},\text{wedges}}(\varepsilon_m^m) &= \tilde{\mathcal{V}}_{m,\Gamma_{W_m},\text{upper wedges}}(\varepsilon_m^m) + \tilde{\mathcal{V}}_{m,\Gamma_{W_m},\text{lower wedges}}(\varepsilon_m^m) + \tilde{\mathcal{V}}_{m,\Gamma_{W_m},\text{extreme wedges}}(\varepsilon_m^m) \\
&= \frac{r_b \pi (\varepsilon_m^m)^3}{8} N_b^{-\{x\}} - \frac{\pi (\varepsilon_m^m)^4}{2} + \pi (\varepsilon_m^m)^2 \\
&\quad - \frac{(\varepsilon_m^m)^3}{4} r_b \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} N_b^{-((2k+1)D_W-2k)\{x\}} (\varepsilon_m^m)^{2k+1} \mathcal{O}(1) \\
&\quad + \frac{(\varepsilon_m^m)^4}{2} \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} N_b^{-((2k+1)D_W-2k+1)\{x\}} (\varepsilon_m^m)^{2k+1} \mathcal{O}(1) \\
&= \frac{r_b \pi}{8} \frac{N_b-1}{N_b} \sum_{\ell \in \mathbb{Z}} \frac{(N_b-1)^{-i\ell \mathbf{p}} (\varepsilon_m^m)^{3-i\ell \mathbf{p}}}{\ln N_b + 2i\ell \pi} - \frac{\pi (\varepsilon_m^m)^4}{2} + \pi (\varepsilon_m^m)^2 \\
&\quad - \frac{1}{4} r_b \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \frac{N_b^{((2k+1)D_W-2k)} - 1}{N_b^{((2k+1)D_W-2k)}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{((2k+1)D_W-2k)} - 1)^{-i\ell \mathbf{p}} (\varepsilon_m^m)^{2k+1-i\ell \mathbf{p}}}{((2k+1)D_W-2k) \ln N_b + 2i\ell \pi} \mathcal{O}(1) \\
&\quad + \frac{1}{2} \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \frac{N_b^{(2k+1)(D_W-1)} - 1}{N_b^{(2k+1)(D_W-1)}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{(2k+1)(D_W-2k+1)} - 1)^{-i\ell \mathbf{p}} (\varepsilon_m^m)^{5+2k-i\ell \mathbf{p}}}{((2k+1)D_W-2k+1) \ln N_b + 2i\ell \pi} \mathcal{O}(1) \tag{R 46}
\end{align*}
$$

Recall from Subsection 4.1 that

$$
\tilde{\mathcal{V}}_{m,\Gamma_{W_m},\star,\text{wedges}}(\varepsilon_m^m) = \mathcal{V}_{m,\Gamma_{W_m},\star,\text{wedges}}(\varepsilon_m^m),
$$

where $\star$ = upper, lower, or extreme. Hence, in light of the first equality in relation (R46), an anal-
ogous identity holds if $\star$, wedges is replaced by “wedges”.

As before, for the sake of clarity, we will rewrite relation (R46) in the form

$$
\begin{align*}
\tilde{\mathcal{V}}_{m,\Gamma_{W_m},\text{wedges}}(\varepsilon_m^m) &= C_{\text{wedges}}^1 \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i \ell \mathbf{p}} (\varepsilon_m^m)^{3-i \ell \mathbf{p}}}{\ln N_b + 2i \ell \pi} - \frac{\pi (\varepsilon_m^m)^4}{2} + \pi (\varepsilon_m^m)^2 \\
&\quad - C_{\text{wedges}}^2 \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \frac{N_b^{((2k+1)D_W-2k)}}{N_b^{((2k+1)D_W-2k)}} - 1 \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{((2k+1)D_W-2k)} - 1)^{-i \ell \mathbf{p}} (\varepsilon_m^m)^{2k+1-i \ell \mathbf{p}}}{((2k+1)D_W-2k) \ln N_b + 2i \ell \pi} \\
&\quad + C_{\text{wedges}}^3 \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \frac{N_b^{(2k+1)(D_W-1)}}{N_b^{(2k+1)(D_W-1)}} - 1 \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{(2k+1)D_W-2k+1)} - 1)^{-i \ell \mathbf{p}} (\varepsilon_m^m)^{5+2k-i \ell \mathbf{p}}}{((2k+1)D_W-2k+1) \ln N_b + 2i \ell \pi}, \tag{R 47}
\end{align*}
$$
### A.4. Proof of Property 2.19, stated on page 27

**Proof.** i. Due to the symmetry with respect to the vertical line $x = \frac{1}{2}$ (see Property 2.1, on page 10), one can restrict oneself to the vertices $P_j$, with $0 < j < \frac{N_b - 1}{2}$.

The initial polygon $\mathcal{P}_0$, admits reentrant interior angles at a vertex $P_j$, with $j + 1 < \frac{N_b - 1}{2}$, in the case when

$$((y'y), (\widehat{P_{j-1}P_j})) > ((y'y), (\widehat{P_jP_{j+1}})). \qquad (59)$$

Therefore, condition (59) above corresponds to the case when

$$|\mathcal{W}\left(\frac{j+1}{N_b-1}\right) - \mathcal{W}\left(\frac{j}{N_b-1}\right)| > |\mathcal{W}\left(\frac{j}{N_b-1}\right) - \mathcal{W}\left(\frac{j-1}{N_b-1}\right)|,$$

i.e.,

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

or, equivalently,

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

and thus happens if

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

Since

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

we conclude that condition (59), on page 68, occurs if

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

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 $\mathcal{P}_0$, admits **reentrant interior angles** at a vertex $P_j$ in the case when $\frac{3N_b - 1}{4} \le j < N_b - 1$.
where $C_{\text{wedges}}^1$, $C_{\text{wedges}}^2$, and $C_{\text{wedges}}^3$ denote strictly positive and finite constants depending on $m$, but uniformly bounded away from 0 and $\infty$ (see Proposition 3.6, on page 57).

The associated (local) effective tube zeta function (see Definition 4.3, on page 62 above) is first obtained, for any complex number $s$ such that $\text{Re}(s) > D_W$, as follows:

$$
\begin{align*}
\tilde{\zeta}_{m, \text{wedges}}^e(s) &= \int_0^{\varepsilon_m^m} t^{s-3} \tilde{\Upsilon}_{m, \Gamma_{W_m}, \text{wedges}}(t) dt \\
&= C_{\text{wedges}}^1 \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i \ell \mathbf{p}}}{\ln N_b + 2 i \ell \pi} \int_0^{\varepsilon_m^m} t^{s-i \ell \mathbf{p}} dt + \pi \int_0^{\varepsilon_m^m} t^{s-1} dt - \frac{\pi}{2} \int_0^{\varepsilon_m^m} t^{s+1} dt \\
&\quad - C_{\text{wedges}}^2 \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \frac{N_b^{((2k+1)D_W-2k)-1}}{N_b^{((2k+1)D_W-2k)}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{((2k+1)D_W-2k)-1})^{-i \ell \mathbf{p}}}{((2k+1)D_W-2k) \ln N_b + 2i\ell\pi} \int_0^{\varepsilon_m^m} t^{s+2k+1-i\ell \mathbf{p}} dt \\
&\quad + C_{\text{wedges}}^3 \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \frac{(-1)^k}{2k+1} \frac{N_b^{(2k+1)(D_W-2k+1)-1}}{N_b^{(2k+1)(D_W-2k+1)}} \frac{(N_b^{(2k+1)D_W-2k+1}-1)^{-i \ell \mathbf{p}}}{((2k+1)D_W-2k+1) \ln N_b + 2i\ell\pi} \int_0^{\varepsilon_m^m} t^{s-2+2k-i\ell \mathbf{p}} dt \\
&= C_{\text{wedges}}^1 \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i \ell \mathbf{p}}}{\ln N_b + 2i\ell\pi} \frac{(\varepsilon_m^m)^{s+1-i \ell \mathbf{p}}}{s+1-i \ell \mathbf{p}} + \frac{\pi (\varepsilon_m^m)^s}{s} - \frac{\pi (\varepsilon_m^m)^{s+2}}{2(s+2)} \\
&\quad - C_{\text{wedges}}^2 \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \frac{N_b^{((2k+1)D_W-2k)-1}}{N_b^{((2k+1)D_W-2k)}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{((2k+1)D_W-2k)-1})^{-i \ell \mathbf{p}}}{((2k+1)D_W-2k) \ln N_b + 2i\ell\pi} \frac{(\varepsilon_m^m)^{s+2k-1-i \ell \mathbf{p}}}{s+2k-1-i \ell \mathbf{p}} \\
&\quad + C_{\text{wedges}}^3 \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \frac{N_b^{(2k+1)(D_W-2k+1)-1}}{N_b^{(2k+1)(D_W-2k+1)}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{(2k+1)D_W-2k+1}-1)^{-i \ell \mathbf{p}}}{((2k+1)D_W-2k+1) \ln N_b + 2i\ell\pi} \frac{(\varepsilon_m^m)^{s+3+2k-i \ell \mathbf{p}}}{s+3+2k-i \ell \mathbf{p}}.
\end{align*}
\tag{R 48}
$$

By meromorphic continuation to all of $\mathbb{C}$, one then obtains $\tilde{\zeta}_{m, \text{wedges}}^e$, the (local) effective tube zeta function (associated with the wedges), for all $s \in \mathbb{C}$, as given by the last two equalities in relation (R48) just above.

The associated Complex Dimensions arise as

$$ -1 + i \ell \mathbf{p}, \quad 1 - 2k + i \ell \mathbf{p}, \quad -3 - 2k + i \ell \mathbf{p}, \quad \text{with } k \in \mathbb{N}, \ell \in \mathbb{Z}, \text{ along with } 0 \text{ and } -2. $$

Note that for $k \ge 2$ (and any $\ell \in \mathbb{Z}$), the last two families of (possible) Complex Dimensions fully overlap. We will take this fact into account in Theorem 4.10, on page 81, and Theorem 4.11, on page 83 below.
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 2.11, on page 21 (given in Appendix A.2, on page 63), 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

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

Thus,

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

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

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

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

$$ (x_k(N_b-1)+j, m-1, W(x_k(N_b-1)+j, m-1)) \quad , \quad \text{with } 0 \le k \le N_b^{m-1}-1. $$
**Property 4.5 (Tube Formula and Effective Tube Zeta Function Associated to the Contribution of the Extra Outer Triangles to the Tubular Volume).**

Given $m \in \mathbb{N}^*$ sufficiently large, with Notation 11, on page 64, we obtain, for the negative contribution of the extra outer triangles to the effective tubular volume $\tilde{V}_{m, \Gamma_{W_m}} (\varepsilon_m^m)$,

$$ 
\begin{align*}
\tilde{V}_{m, \Gamma_{W_m}, \text{extra outer triangles}} (\varepsilon_m^m) &= \tilde{V}_{m, \Gamma_{W_m}, \text{extra outer lower triangles}} (\varepsilon_m^m) + \tilde{V}_{m, \Gamma_{W_m}, \text{extra outer upper triangles}} (\varepsilon_m^m) \\
&= -\frac{(\varepsilon_m^m)^2}{2} N_b^{(3-D_W)\{x\}} \mathcal{O}(1) \\
&= -\frac{N_b^{D_W-3} - 1}{N_b^{D_W-3}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{D_W-3} - 1)^{-i\ell \mathbf{p}} (\varepsilon_m^m)^{2-i\ell \mathbf{p}}}{(D_W-3) \ln N_b + 2i\ell\pi} \mathcal{O}(1), \tag{R 49}
\end{align*}
$$

with

$$0 < C_{\text{inf}}^3 \le \mathcal{O}(1) \le C_{\text{sup}}^3 < \infty.$$

Recall from Subsection 4.1 that $\tilde{V}_{m, \Gamma_{W_m}, \star} (\varepsilon_m^m) = V_{m, \Gamma_{W_m}, \star} (\varepsilon_m^m)$, where $\star = \text{extra outer lower triangles}$, or extra outer upper triangles. Hence, in light of the first equality in relation (R49), we also have that $\tilde{V}_{m, \Gamma_{W_m}, \text{extra outer triangles}} (\varepsilon_m^m) = V_{m, \Gamma_{W_m}, \text{extra outer triangles}} (\varepsilon_m^m)$.

As previously, for the sake of clarity, we will write relation (R49) in the following form:

$$ \tilde{V}_{m, \Gamma_{W_m}, \text{extra outer triangles}} (\varepsilon_m^m) = -C_{\text{triangles}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{D_W-3} - 1)^{-i \ell \mathbf{p}} (\varepsilon_m^m)^{2-i \ell \mathbf{p}}}{(D_W-3) \ln N_b + 2i \ell \pi}, \quad (\mathbb{R} 50) $$

where $C_{\text{triangles}}$ denotes a strictly positive and finite constant, depending on $m$, but uniformly bounded away from 0 and $\infty$ (in $m \in \mathbb{N}^*$ sufficiently large); see Proposition 3.7, on page 58. More specifically,

$$0 < C_{\text{inf}}^3 \le C_{\text{triangles}} \le C_{\text{sup}}^3 < \infty.$$

The associated (local) effective tube zeta function (see Definition 4.3, on page 62 above) is first obtained, for any complex number $s$ such that $\text{Re}(s) > D_W$, as follows:

$$ 
\begin{align*}
\tilde{\zeta}_{m, \text{extra outer triangles}}^e(s) &= \int_0^{\varepsilon_m^m} t^{s-3} \tilde{V}_{m, \Gamma_{W_m}, \text{extra outer triangles}}(t) dt \\
&= -C_{\text{triangles}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{D_W-3} - 1)^{-i \ell \mathbf{p}}}{(2 - 3 D_W) \ln N_b + 2i \ell \pi} \int_0^{\varepsilon_m^m} t^{s-2-i \ell \mathbf{p}} dt \\
&= -C_{\text{triangles}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{D_W-3} - 1)^{-i \ell \mathbf{p}}}{(D_W - 3) \ln N_b + 2i \ell \pi} \frac{(\varepsilon_m^m)^{s-1-i \ell \mathbf{p}}}{s-1-i \ell \mathbf{p}}. \tag{R 51}
\end{align*}
$$

By meromorphic continuation to all of $\mathbb{C}$, one then obtains $\tilde{\zeta}_{m, \text{extra outer triangles}}^e$, the (local) effective tube zeta function (associated with the extra outer triangles), for all $s \in \mathbb{C}$, as given by the last two
We then want to prove that there is a reentrant interior angle at any vertex

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

As was the case in the proof of Property 2.11 (stated on page 21), given in Appendix A.2, on page 63, 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:

$$ k = \tilde{k} + i N_b^{m-1} , \quad 0 \le \tilde{k} \le N_b^{m-1} - 1 , \quad 0 \le i \le N_b - 1 . \quad (60) $$

Thus,

$$ \begin{aligned} W(x_k(N_b-1)+j+1,m) - W(x_{j,m}) & = \lambda \left( W(x_{\tilde{k}(N_b-1)+j+1,m-1}) - W(x_{\tilde{k}(N_b-1)+j,m-1}) \right) \\ & \quad -2 \sin \left( \frac{\pi}{(N_b-1) N_b^{m-1}} \right) \sin \left( \frac{(2(N_b-1)\tilde{k} + 2j+1)\pi}{(N_b-1) N_b^{m-1}} \right), \end{aligned} \quad (61) $$

and

$$ \begin{aligned} W(x_k(N_b-1)+j,m) - W(x_{j-1,m}) & = \lambda \left( W(x_{\tilde{k}(N_b-1)+j,m-1}) - W(x_{\tilde{k}(N_b-1)+j-1,m-1}) \right) \\ & \quad -2 \sin \left( \frac{\pi}{(N_b-1) N_b^{m-1}} \right) \sin \left( \frac{(2(N_b-1)\tilde{k} + 2j-1)\pi}{(N_b-1) N_b^{m-1}} \right). \end{aligned} \quad (62) $$

In light of Property 2.11, on page 21, 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

$$ \begin{aligned} W(x_{\tilde{k}(N_b-1)+j-1,m-1}) - W(x_{j,m-1}) &> 0 , \\ W(x_{\tilde{k}(N_b-1)+j,m-1}) - W(x_{\tilde{k}(N_b-1)+j+1,m-1}) &> 0 . \end{aligned} $$
equalities in relation (R51) just above.

The associated Complex Dimensions arise as

$$1 + i \ell \mathbf{p}, \text{ with } \ell \in \mathbb{Z}.$$

**Property 4.6 (Tube Formula and Effective Tube Zeta Function Associated to the Contribution of the Parallelograms to the Tubular Volume).**

Given $m \in \mathbb{N}^*$ sufficiently large, with Notation 11, on page 64, the last contribution to the effective tubular volume $\tilde{V}_{m, \Gamma_{W_m}} (\varepsilon_m^m)$, coming from the parallelograms, is given by

$$\tilde{V}_{m, \Gamma_{W_m}, \text{parallelograms}} (\varepsilon_m^m) = \tilde{V}_{m, \Gamma_{W_m}, \text{lower parallelograms}} (\varepsilon_m^m) + \tilde{V}_{m, \Gamma_{W_m}, \text{upper parallelograms}} (\varepsilon_m^m)$$

$$= -C_{\text{parallelograms}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{D_W-3} - 1)^{-i \ell \mathbf{p}} (\varepsilon_m^m)^{2-i \ell \mathbf{p}}}{(D_W - 3) \ln N_b + 2 i \ell \pi},$$

(R52)

where $C_{\text{parallelograms}}$ denotes a strictly positive and finite constant, depending on $m$, but uniformly bounded away from 0 and $\infty$ (see Proposition 3.8, on page 59). More specifically, again,

$$0 < C_{\text{inf}}^3 \le C_{\text{parallelograms}} \le C_{\text{sup}}^3 < \infty.$$

Also, recall from Subsection 4.1 that, by construction,

$$\tilde{V}_{m, \Gamma_{W_m}, \text{lower parallelograms}} (\varepsilon_m^m) = V_{m, \Gamma_{W_m}, \text{lower parallelograms}} (\varepsilon_m^m),$$

and similarly, if “lower parallelograms” is replaced by “upper parallelograms”. Hence, an entirely analogous relation holds if “parallelograms” is substituted for “lower parallelograms”.

The associated (local) effective tube zeta function $\tilde{\zeta}_{m, \text{parallelograms}}^e$ (see Definition 4.3, on page 62 above) is then first obtained, for any complex number $s$ such that $\text{Re}(s) > D_W$, as follows:

$$\begin{align*}
\tilde{\zeta}_{m, \text{parallelograms}}^e(s) &= \int_0^{\varepsilon_m^m} t^{s-3} \tilde{V}_{m, \Gamma_{W_m}, \text{parallelograms}}(t) dt \\
&= -C_{\text{parallelograms}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{2-3D_W} - 1)^{-i\ell \mathbf{p}}}{(2-3D_W) \ln N_b + 2i\ell\pi} \int_0^{\varepsilon_m^m} t^{s-2-i\ell \mathbf{p}} dt \\
&= -C_{\text{parallelograms}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{D_W-3} - 1)^{-i\ell \mathbf{p}}}{(D_W - 3) \ln N_b + 2i\ell\pi} \frac{(\varepsilon_m^m)^{s-1-i\ell \mathbf{p}}}{s-1-i\ell \mathbf{p}}.
\end{align*}$$

(R53)

By meromorphic continuation to all of $\mathbb{C}$, one then obtains $\tilde{\zeta}_{m, \text{parallelograms}}^e$, the (local) effective tube zeta function (associated with the parallelograms), for all $s \in \mathbb{C}$, as given by the last two equalities in relation (R53) just above.

The associated Complex Dimensions arise as

$$1 + i \ell \mathbf{p}, \text{ with } \ell \in \mathbb{Z}.$$
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.,

$$ \frac{L_{m-1}}{|W(x_{\tilde{k}(N_b-1)+j,m-1}) - W(x_{\tilde{k}(N_b-1)+j-1,m-1})|} > \frac{L_{m-1}}{|W(x_{\tilde{k}(N_b-1)+j+1,m-1}) - W(x_{\tilde{k}(N_b-1)+j,m-1})|}, $$

which yields

$$ \begin{aligned} & |W(x_{\tilde{k}(N_b-1)+j+1,m-1}) - W(x_{\tilde{k}(N_b-1)+j,m-1})| \\ & > |W(x_{\tilde{k}(N_b-1)+j,m-1}) - W(x_{\tilde{k}(N_b-1)+j-1,m-1})|, \end{aligned} $$

or, equivalently,

$$ \begin{aligned} & W(x_{\tilde{k}(N_b-1)+j,m-1}) - W(x_{\tilde{k}(N_b-1)+j+1,m-1}) \\ & > W(x_{\tilde{k}(N_b-1)+j-1,m-1}) - W(x_{\tilde{k}(N_b-1)+j,m-1}). \end{aligned} $$

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

$$ (x_{(N_b-1)\tilde{k}+j,m-2}, W(x_{(N_b-1)\tilde{k}+j,m-2})), $$

requires, in conjunction with

$$ \begin{aligned} & W(x_{\tilde{k}(N_b-1)+j,m-2}) - W(x_{\tilde{k}(N_b-1)+j+1,m-2}) \\ & > W(x_{\tilde{k}(N_b-1)+j-1,m-2}) - W(x_{\tilde{k}(N_b-1)+j,m-2}), \end{aligned} $$

that

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

which corresponds to

$$ 0 < \frac{\pi (2 \tilde{k} (N_b - 1) + 2 j + 1)}{(N_b - 1) N_b^{m-2}} < \frac{\pi (2 \tilde{k} (N_b - 1) + 2 j - 1)}{(N_b - 1) N_b^{m-2}} \le \frac{\pi}{2} $$
The above results stated in Properties 4.3–4.6, on pages 65–71, can now be combined in order to
yield the following key theorems:

**Theorem 4.7 (Fractal Tube Formula for The Weierstrass IFD).**

Given $m \in \mathbb{N}$ sufficiently large, the $m^{th}$ total contribution to the tubular volume $V_{m,\Gamma_{W_m}}(\varepsilon_m^m)$, or two-dimensional Lebesgue measure of the $\varepsilon_m^m$-neighborhood of the $m^{th}$ prefractal approximation $\Gamma_{W_m}$,

$$
\mathcal{D}(\varepsilon_m^m) = \{M = (x, y) \in \mathbb{R}^2, d(M, \Gamma_{W_m}) \le \varepsilon_m^m\}, \quad (\mathbb{R}^54)
$$

where $\varepsilon = (\varepsilon_m^m)_{m \in \mathbb{N}}$ is the cohomology infinitesimal, as introduced in Definition 3.1, on page 37, is given by

$$
\tilde{V}_{m,\Gamma_{W_m}}(\varepsilon_m^m) = \tilde{V}_{m,\Gamma_{W_m},\text{Rectangles}}(\varepsilon_m^m) + \tilde{V}_{m,\Gamma_{W_m},\text{wedges}}(\varepsilon_m^m) \\
+ \tilde{V}_{m,\Gamma_{W_m},\text{extra outer triangles}}(\varepsilon_m^m) + \tilde{V}_{m,\Gamma_{W_m},\text{parallelograms}}(\varepsilon_m^m), \tag{R 55}
$$

i.e.,

$$
\begin{align*}
\tilde{V}_{m,\Gamma_{W_m}}(\varepsilon_m^m) ={}& C_{\text{Rectangles}} \sum_{k=0}^{\infty} \left(\frac{1}{k}\right) \frac{N_b^{1-k(2-D_W)} - 1}{N_b^{1-k(2-D_W)}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i \ell \mathbf{p}}}{(1-k(2-D_W)) \ln N_b + 2i\ell\pi} (\varepsilon_m^m)^{2-D_W+k(2-D_W)-i\ell\mathbf{p}} \\
& + C_{\text{wedges}}^1 \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i \ell \mathbf{p}} (\varepsilon_m^m)^{3-i \ell \mathbf{p}}}{\ln N_b + 2i\ell\pi} + \pi (\varepsilon_m^m)^2 - \frac{\pi (\varepsilon_m^m)^4}{2} \\
& - C_{\text{wedges}}^2 \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \frac{N_b^{((2k+1)D_W-2k)} - 1}{N_b^{((2k+1)D_W-2k)}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{((2k+1)D_W-2k)} - 1)^{-i \ell \mathbf{p}} (\varepsilon_m^m)^{2k+1-i \ell \mathbf{p}}}{((2k+1)D_W-2k) \ln N_b + 2i\ell\pi} \\
& + C_{\text{wedges}}^3 \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \frac{N_b^{(2k+1)(D_W-1)} - 1}{N_b^{(2k+1)(D_W-1)}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{(2k+1)(D_W-1)} - 1)^{-i \ell \mathbf{p}} (\varepsilon_m^m)^{5+2k-i \ell \mathbf{p}}}{((2k+1)D_W-2k+1) \ln N_b + 2i\ell\pi} \\
& - (C_{\text{triangles}} + C_{\text{parallelograms}}) \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{2-3D_W} - 1)^{-i \ell \mathbf{p}}}{(2-3D_W) \ln N_b + 2i\ell\pi} (\varepsilon_m^m)^{2-i \ell \mathbf{p}}, \tag{R 56}
\end{align*}
$$

where $C_{\text{rectangles}}$, $C_{\text{wedges}}^{\ell}$, $\ell = 1, 2, 3$, $C_{\text{triangles}}$, and $C_{\text{parallelograms}}$ denote the strictly positive and finite constants respectively introduced in Properties 4.3–4.6, on pages 65–71 above. Recall that these constants depend on $m$, but are uniformly bounded away from $0$ and $\infty$ (in $m \in \mathbb{N}^*$ large enough).

Also, recall from the discussion in Subsection 4.1 that, by construction,

$$
\tilde{V}_{m,\Gamma_{W_m}}(\varepsilon_m^m) = V_{m,\Gamma_{W_m}}(\varepsilon_m^m) .
$$

Actually, this identity follows from the corresponding identity for each of the terms on the right-
hand side of relation (R55).
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:

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

Hence,

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

||

$$\frac{L_m}{|W(x_{\tilde{k}}(N_b-1)+j,m) - W(x_{\tilde{k}}(N_b-1)+j-1,m)|}$$

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

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

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

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

which yields the expected result. Namely,

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

i.e., the presence of a reentrant angle at the $j^{th}$ vertex of the polygon $P_{m,k}$.
For the sake of clarity, and in order to highlight the role played by the one-periodic functions (with respect to the variable $\ln N_b (\epsilon_m^m)^{-1}$, see Property 4.1, on page 61), one can exchange the sums over $k$ and $m$, which enables one to obtain an expression of the following form:

$$ \tilde{V}_{m, \Gamma_{Wm}} (\epsilon_m^m) = \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}} f_{k, \ell, \text{Rectangles}} (\epsilon_m^m)^{2-D_W+k(2-D_W)-i\ell\mathbf{p}} \\ + \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}} (f_{k, \ell, \text{wedges},1} (\epsilon_m^m)^{3-i\ell\mathbf{p}} + f_{k, \ell, \text{wedges},2} (\epsilon_m^m)^{1+2k-i\ell\mathbf{p}} + f_{k, \ell, \text{wedges},3} (\epsilon_m^m)^{5+2k-i\ell\mathbf{p}}) \\ + \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}} f_{k, \ell, \text{triangles, parallelograms}} (\epsilon_m^m)^{2-i\ell\mathbf{p}} + \pi (\epsilon_m^m)^2 - \frac{\pi (\epsilon_m^m)^4}{2}, \quad (\mathbb{R} 57) $$

where the notation $f_{k,\ell,\text{Rectangles}}$, $f_{k,\ell,\text{wedges},\ell'}$, $1 \le \ell' \le 3$, and $f_{k,\ell,\text{triangles, parallelograms}}$, respectively account for the nonzero coefficients associated to the sums corresponding to the contribution of the rectangles, wedges, triangles and parallelograms, respectively given by:

$$ f_{k,\ell,\text{Rectangles}} = C_{\text{Rectangles}} \left(\frac{1}{k}\right) \frac{N_b^{1-k(2-D_W)} - 1}{N_b^{1-k(2-D_W)}} \frac{(N_b - 1)^{-i\ell\mathbf{p}}}{(1 - k(2 - D_W)) \ln N_b + 2i\ell\pi}; \quad (\mathbb{R} 58) $$

$$ f_{k,\ell,\text{wedges},1} = C_{\text{wedges}}^1 \frac{(N_b - 1)^{-i\ell\mathbf{p}}}{\ln N_b + 2i\ell\pi}; \quad (\mathbb{R} 59) $$

$$ f_{k,\ell,\text{wedges},2} = -C_{\text{wedges}}^2 \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \frac{N_b^{((2k+1)D_W-2k)-1}}{N_b^{((2k+1)D_W-2k)}} \frac{(N_b^{((2k+1)D_W-2k)-1})^{-i\ell\mathbf{p}}}{((2k+1)D_W-2k) \ln N_b + 2i\ell\pi}; \quad (\mathbb{R} 60) $$

$$ f_{k,\ell,\text{wedges},3} = C_{\text{wedges}}^3 \frac{(-1)^k}{2k+1} \frac{N_b^{(2k+1)(D_W-1)-1}}{N_b^{(2k+1)(D_W-1)}} \frac{(N_b^{(2k+1)(D_W-2k+1)-1})^{-i\ell\mathbf{p}}}{((2k+1)D_W-2k+1) \ln N_b + 2i\ell\pi}; \quad (\mathbb{R} 61) $$

$$ f_{k,\ell,\text{triangles, parallelograms}} = -(C_{\text{triangles}} + C_{\text{parallelograms}}) \frac{(N_b^{2-3D_W}-1)^{-i\ell\mathbf{p}}}{(2-3D_W)\ln N_b + 2i\ell\pi}. \quad (\mathbb{R} 62) $$

Note that those coefficients do not depend on $\epsilon_m^m$, and satisfy the following uniform estimates (independent of $m \in \mathbb{N}^*$ sufficiently large):

$$ |f_{k,\ell,\text{Rectangles}}| \le C_{\text{Rectangles}} \left(\frac{1}{k}\right) \frac{1}{2\ell\pi}; \quad (\mathbb{R} 63) $$

$$ |f_{k,\ell,\text{wedges},1}| \le \frac{C_{\text{wedges}}^1}{2\ell\pi}; \quad (\mathbb{R} 64) $$

$$ |f_{k,\ell,\text{wedges},2}| \le \frac{C_{\text{wedges}}^2}{2k+1} \frac{1}{2\ell\pi}; \quad (\mathbb{R} 65) $$

$$ |f_{k,\ell,\text{wedges},3}| \le \frac{C_{\text{wedges}}^3}{2k+1} \frac{1}{2\ell\pi}; \quad (\mathbb{R} 66) $$

$$ |f_{k,\ell,\text{triangles, parallelograms}}| \le (C_{\text{triangles}} + C_{\text{parallelograms}}) \cdot \quad (\mathbb{R} 67) $$

Finally, each of the double sums in formulae (R55), on page 72, and (R57), on page 73, is absolutely convergent (and hence, convergent).
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

$$ \theta_k(N_b-1)+j-1, \tilde{k}(N_b-1)+j, mc < \theta_k(N_b-1)+j, \tilde{k}(N_b-1)+j+1, m $$

and

$$ \begin{aligned} W(x_k(N_b-1)+j-1, m) - W(x_k(N_b-1)+j, m) &< 0, \\ W(x_k(N_b-1)+j, m) - W(x_k(N_b-1)+j+1, m) &< 0. \end{aligned} $$

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 2.19, stated on page 27. □

### A.5. Proof of Corollary 3.3, stated on page 36

**Proof.** We simply use Theorem 3.2, on page 33. Since, for any sufficiently large positive integer $m$ and any $j$ in $\{1, \dots, \#V_m - 1\}$,

$$ h_{j-1,m} = W(j \varepsilon_m^m) - W((j-1) \varepsilon_m^m), $$

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

$$ \begin{aligned} h_{j-1,m} &= W(j \varepsilon_m^m) - W((j-1) \varepsilon_m^m) \\ &= \sum_{k=0}^{m} \varepsilon^{k(2-D_W)} (c_{k,j,m} \varepsilon^{i \ell_{j_k,m,k}} \mathbf{P} - c_{k,j-1,m} \varepsilon^{i \ell_{j_{k-1,m,k}}}) \end{aligned} \quad (63) $$

and, in the case when $h_{j-1,m} \le 0$,

$$ \begin{aligned} h_{j-1,m} &= -W(j \varepsilon_m^m) + W((j-1) \varepsilon_m^m) \\ &= \sum_{k=0}^{m} \varepsilon^{k(2-D_W)} (c_{k,j-1,m} \varepsilon^{i \ell_{j_{k-1,m,k}}} \mathbf{P} - c_{k,j,m} \varepsilon^{i \ell_{j_{k,m,k}}} \mathbf{P}). \end{aligned} \quad (64) $$

This amounts to writing $h_{j-1,m}$ in the following form,

$$ h_{j-1,m} = \sum_{k=0}^{m} d_{k,j,m} \varepsilon^{k(2-D_W)} \varepsilon^{i \ell_{k,j,m,k}} \mathbf{P}, \quad (65) $$
Proof. Indeed, by construction, the identity $(\mathbb{R}55)$, on page 72, holds. Therefore, all of the main statements in the theorem concerning the $m^{th}$ effective tubular volume $\tilde{\mathcal{V}}_{m,\Gamma_{\mathcal{W}}m} (\varepsilon_m^m)$ follow by combining Properties 4.3–4.6, on pages 65–71 above.

Finally, we justify the uniform estimates $(\mathbb{R}63)–(\mathbb{R}67)$ in the following manner:

We have that

$$
\begin{align*}
|f_{k,l,\text{Rectangles}}| &\le C_{\text{Rectangles}} \binom{\frac{1}{2}}{k} \frac{|N_b^{1-k}(2-D_W) - 1|}{N_b^{1-k}(2-D_W)} \frac{1}{\sqrt{(1-k(2-D_W))^2 (\ln N_b)^2 + 4 \ell^2 \pi^2}} \\
&\le C_{\text{Rectangles}} \binom{\frac{1}{2}}{k} \frac{1}{\sqrt{(1-k(2-D_W))^2 (\ln N_b)^2 + 4 \ell^2 \pi^2}} \\
&\le C_{\text{Rectangles}} \binom{\frac{1}{2}}{k} \frac{1}{2 \ell \pi};
\end{align*}
$$

$$
|f_{k,l,\text{wedges},1}| \le C_{\text{wedges}}^1 \frac{1}{\sqrt{(\ln N_b)^2 + 4 \ell^2 \pi^2}} \le \frac{C_{\text{wedges}}^1}{2 \ell \pi};
$$

$$
\begin{align*}
|f_{k,l,\text{wedges},2}| &\le \frac{C_{\text{wedges}}^2}{2k+1} \left| \frac{N_b^{((2k+1)D_W-2k)} - 1}{N_b^{((2k+1)D_W-2k)}} \right| \frac{1}{\sqrt{((2k+1)D_W-2k)^2 (\ln N_b)^2 + 4\ell^2\pi^2}} \\
&\le \frac{C_{\text{wedges}}^2}{2k+1} \frac{1}{\sqrt{((2k+1)D_W-2k)^2 (\ln N_b)^2 + 4\ell^2\pi^2}} \\
&\le \frac{C_{\text{wedges}}^2}{2k+1} \frac{1}{2\ell\pi};
\end{align*}
$$

$$
\begin{align*}
|f_{k,l,\text{wedges},3}| &\le \frac{C_{\text{wedges}}^3}{2k+1} \left| \frac{N_b^{(2k+1)(D_W-1)} - 1}{N_b^{(2k+1)(D_W-1)}} \right| \frac{1}{\sqrt{((2k+1)D_W - 2k+1)^2 (\ln N_b)^2 + 4\ell^2\pi^2}} \\
&\le \frac{C_{\text{wedges}}^3}{2k+1} \frac{1}{\sqrt{((2k+1)D_W - 2k+1)^2 (\ln N_b)^2 + 4\ell^2\pi^2}} \\
&\le \frac{C_{\text{wedges}}^3}{2k+1} \frac{1}{2\ell\pi};
\end{align*}
$$

$$
\begin{align*}
|f_{k,l,\text{triangles, parallelograms}}| &\le (C_{\text{triangles}} + C_{\text{parallelograms}}) \frac{1}{\sqrt{(2-3D_W)^2 (\ln N_b)^2 + 4\ell^2\pi^2}} \\
&\le (C_{\text{triangles}} + C_{\text{parallelograms}}) \frac{1}{2\ell\pi}.
\end{align*}
$$
with $\ell_{k,j,m} \in \mathbb{Z}$ and where:

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

$$ 
\begin{aligned}
d_{k,j,m} \varepsilon^{i \ell_{k,j,m}} &= c_{k,j,m} \varepsilon^{i \ell_{j,k,m,k}} \mathbf{P} - c_{k,j-1,m} \varepsilon^{i \ell_{j,k-1,m,k}} \mathbf{P} \\
&= \varepsilon^{i \ell_{j,k,m,k}} \mathbf{P} \left( c_{k,j,m} - c_{k,j-1,m} \varepsilon^{i (\ell_{j,k-1,m,k} - \ell_{j,k,m,k})} \mathbf{P} \right),
\end{aligned}
$$

which yields

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

and

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

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

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

which yields

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

and

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

### A.6. Proof of Theorem 4.3, stated on page 54

**Proof.** Since, by definition (see relation (38) in Definition 4.1, on page 51),

$$ \tilde{\zeta}_m^e(s) = \int_0^s t^{s-3} \tilde{V}_m(t) \, dt, \qquad (66) $$

for all $m \in \mathbb{N}^*$ sufficiently large, and for all $s \in \mathbb{C}$ with $\mathcal{R}e(s)$ sufficiently large, we have that

$$ 
\begin{aligned}
\tilde{\zeta}_m^e(s) = & \sum_{j=1}^{\#V_m-1} \sum_{k=0}^{m} \frac{\overline{C_{k,j,m}} \varepsilon^{s-D_W+(k-1)(2-D_W)-i\ell_{k,j,m}} \mathbf{P}}{s - D_W + (k-1)(2-D_W) - i\ell_{k,j,m} \mathbf{P}} \\
& + \sum_{j=1}^{\#V_m-1} \sum_{k=0}^{m} \frac{C_{k,j,m} \varepsilon^{s-D_W+(k-1)(2-D_W)+i\ell_{k,j,m}} \mathbf{P}}{s - D_W + (k-1)(2-D_W) + i\ell_{k,j,m} \mathbf{P}} + \frac{\pi \varepsilon^s}{2(N_b-1)^2 s}.
\end{aligned}
\qquad (67) $$
This concludes the proof of the theorem.

Remark 4.6 (Periodic Functions Associated to the Poles of the Zeta Functions).

The one-periodic functions (with respect to the variable $\ln_{N_b}(\varepsilon_m^m)^{-1}$, see Property 4.1, on page 61), respectively associated to the following values,

i. $D_W - k(2 - D_W)$, with $k \in \mathbb{N}$,

ii. 1,

are nonconstant, since their $m^{th}$ Fourier coefficients, with $m \in \mathbb{Z}$, $m \neq 0$, which are respectively proportional to the strictly positive and finite constants $C_{\text{Rectangles}}$, $C_{\text{wedges}}^{\ell}$, for $1 \le \ell \le 3$, and $C_{\text{triangles}} + C_{\text{parallelograms}}$, are thus nonzero. We refer to Theorem 4.11, on page 83 below, for the specific manner in which these periodic functions arise. (See also Subsection 4.3.2, on page 83.)

## 4.3 Complex Dimensions

We deduce at once the Complex Dimensions of the Weierstrass IFD from the fractal tube formula and the expression for the (local) effective tube zeta function obtained in Theorem 4.7, on page 72 above, and Theorem 4.8, on page 76 below, respectively.

### 4.3.1 Main Results

Following (as well as adapting to IFDs) [LRŽ17b], we hereafter define the local and global effective tube zeta functions of the sequence of Weierstrass IFDs associated to the cohomology infinitesimal, as introduced in Definition 3.1, on page 37.

**Definition 4.4 (Local Tube Zeta Function for the Weierstrass Iterated Fractal Drums).**

In the sequel, for each $m \in \mathbb{N}$, $\tilde{\zeta}_{m,W}^e$ denotes the $m^{th}$ effective tubular zeta function associated with $V_{m,\Gamma_{W_m}}(\varepsilon_m^m)$ – and hence also, associated with the corresponding natural volume extension function $\tilde{V}_{m,\Gamma_{W_m}}(\varepsilon_m^m)$; see Definition 4.3, on page 62, along with Notation 11, on page 64. More specifically, it is initially defined by the following truncated Mellin transform, for all $s \in \mathbb{C}$ with $\text{Re}(s)$ sufficiently large (in fact, for all $s \in \mathbb{C}$ with $\text{Re}(s) > D_W$),

$$ \tilde{\zeta}_{m,\Gamma_{W_m}}^e(s) = \int_0^{\varepsilon_m^m} t^{s-3} \tilde{V}_{m,\Gamma_{W_m}}(t) dt. \qquad (\mathcal{R} 68) $$

We also call $\tilde{\zeta}_{m,\Gamma_{W_m}}^e$ the $m^{th}$ local effective tube zeta function (or the $m^{th}$ prefractal effective tube zeta function) of the Weierstrass IFD, for the same reason as the one provided in Property 4.3, on page 65.
We note that this expression involves the pole $s = 2$, which comes from the Euclidean rectangles. For this reason, as in [14], we hereafter exclude the terms

$$ \frac{\overline{C_{0,j,m}} \varepsilon^{s-D_W-(2-D_W)-i \ell_{0,j,m} \mathbf{p}}}{s-D_W-(2-D_W)-i \ell_{0,j,m} \mathbf{p}} \quad \text{and} \quad \frac{C_{0,j,m} \varepsilon^{s-D_W-(2-D_W)+i \ell_{0,j,m} \mathbf{p}}}{s-D_W-(2-D_W)+i \ell_{0,j,m} \mathbf{p}} $$

from the expression for the $m^{th}$ effective local zeta function $\tilde{\zeta}_m^e$. For the sake of simplicity, we still use the same notation $\tilde{\zeta}_m^e$ for this $m^{th}$ restricted effective local polyhedral zeta function; i.e.,

$$ \tilde{\zeta}_m^e(s) = \sum_{j=1}^{\#V_m-1} \sum_{k=0}^{m} \frac{\overline{C_{k,j,m}} \varepsilon^{s-D_W+(k-1)(2-D_W)-i \ell_{k,j,m} \mathbf{p}}}{s-D_W+(k-1)(2-D_W)-i \ell_{k,j,m} \mathbf{p}} \\ + \sum_{j=1}^{\#V_m-1} \sum_{k=0}^{m} \frac{C_{k,j,m} \varepsilon^{s-D_W+(k-1)(2-D_W)+i \ell_{k,j,m} \mathbf{p}}}{s-D_W+(k-1)(2-D_W)+i \ell_{k,j,m} \mathbf{p}}. \qquad (68) $$

It follows that, for all $m \in \mathbb{N}^*$ sufficiently large, $\tilde{\zeta}_m^e$ has a meromorphic continuation to all of $\mathbb{C}$ given by formula (68) just above.

Finally, the fact that, for all $m$ sufficiently large, the abscissa of convergence of $\tilde{\zeta}_m^e$ coincides with $D_W$ immediately follows by combining formula (68) (for all $s \in \mathbb{C}$) and the method of proof of Theorem 2.1 on page 57 in [39].

The fact that the effective global tube zeta function $\tilde{\zeta}_{\Gamma_W}^e$ admits a meromorphic continuation to all of $\mathbb{C}$ is obtained by applying Weierstrass's Theorem for (locally) uniformly convergent sequences of holomorphic functions. First, for all sufficiently large $m \in \mathbb{N}$, the set $\mathcal{Z}$ of possible poles of the local tube zeta function $\tilde{\zeta}_m^e(s)$ is given by Theorem 4.5, page 56 above. Note that $\mathcal{Z}$ is discrete, and thus closed in $\mathbb{C}$. It then makes sense to consider any of those poles, that we will denote by $\omega$. The local tube zeta function $\tilde{\zeta}_m^e$ is then holomorphic on the connected open subset of $\mathbb{C}$ given by $\mathbb{C} \setminus \mathcal{Z}$. We can clearly see that the sequence of functions $(\tilde{\zeta}_m^e)_{m \ge m_0}$ converges normally (and hence, uniformly) in a connected open (and relatively compact) neighborhood of any given $\omega \in \mathcal{Z}$ - i.e., for $s = x + iy \in \mathbb{C}$ close to $\omega$. Weierstrass's Theorem, applied once again, then ensures the holomorphicity of the limit $\tilde{\zeta}_{\Gamma_W}^e$ on the domain $\mathbb{C} \setminus \mathcal{Z}$. It follows that the global tube zeta function $\tilde{\zeta}_{\Gamma_W}^e$ is meromorphic in all of $\mathbb{C}$, with possible set of poles given by $\mathcal{Z}$.

In closing, we note that the fact that the effective global tube zeta function $\tilde{\zeta}_{\Gamma_W}$ exists, is meromorphic on $\mathbb{C}$, and is given by the limit appearing in relation (43), on page 54, is established in [14]. $\square$

## References

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**Theorem 4.8 (Local and Global Tube Zeta Function for the Weierstrass Iterated Fractal Drums [DL23b]).**

With the notation and terminology of Definition 4.4 just above, $\tilde{\zeta}_{\Gamma_W}^e$, the global effective tube zeta function of the Weierstrass IFD, defined by analogy with the work in [LRŽ17b], admits a (necessarily unique) meromorphic continuation to all of $\mathbb{C}$, and is given, for any $s \in \mathbb{C}$, by the following expression (see [DL23b] for the proof of the existence of the limit, which is locally uniform on $\mathbb{C}$):

$$ \tilde{\zeta}_{\Gamma_W}^e(s) = \lim_{m \to \infty} \tilde{\zeta}_{m, \Gamma_{Wm}}^e(s), \qquad (\mathbb{R} 69) $$

where, for all $m \in \mathbb{N}$ sufficiently large, the $m^{th}$ local effective tube zeta function $\tilde{\zeta}_{m,W}^e$ is given, for any $s \in \mathbb{C}$, by

$$ \begin{aligned} \tilde{\zeta}_{m, \Gamma_{Wm}}^{e}(s) = & \sum_{\ell \in Z, k \in \mathbb{N}} f_{k, \ell, \text{Rectangles}} \frac{(\varepsilon_m^m)^{s-D_W+k(2-D_W)-i \ell \mathbf{p}}}{s - D_W + k(2 - D_W) - i \ell \mathbf{p}} \\ & + \sum_{\ell \in Z, k \in \mathbb{N}} \left\{ f_{\ell, k, \text{wedges}, 1} \frac{(\varepsilon_m^m)^{s+1-i \ell \mathbf{p}}}{s+1-i \ell \mathbf{p}} + f_{\ell, k, \text{wedges}, 2} \frac{(\varepsilon_m^m)^{s+2k-1-i \ell \mathbf{p}}}{s+2k-1-i \ell \mathbf{p}} + f_{\ell, k, \text{wedges}, 3} \frac{(\varepsilon_m^m)^{s+3+2k-i \ell \mathbf{p}}}{s+3+2k-i \ell \mathbf{p}} \right\} \\ & + \sum_{\ell \in Z, k \in \mathbb{N}} f_{k, \ell, \text{triangles, parallelograms}} \frac{(\varepsilon_m^m)^{s-1-i \ell \mathbf{p}}}{s-1-i \ell \mathbf{p}} + \frac{\pi (\varepsilon_m^m)^s}{s} - \frac{\pi (\varepsilon_m^m)^{s+2}}{4(s+2)}, \end{aligned} \qquad (\mathbb{R} 70) $$

where, as already introduced in Theorem 4.7, on page 72, the coefficients $f_{k,\ell,\text{Rectangles}}$, $f_{\ell,k,\text{wedges},j}$, for $1 \le j \le 3$, and $f_{\ell,k,\text{triangles, parallelograms}}$, respectively, depend on $m$, but are uniformly bounded (in $m \in \mathbb{N}^*$ large enough) and account for the nonzero coefficients associated to the sums corresponding to the contribution of the rectangles, wedges, triangles and parallelograms.

Note that, in light of Definition 4.3, on page 62, $\tilde{\zeta}_{m,\Gamma_{Wm}}^e$ is a (tamed) Dirichlet-type integral (in the sense of [LRŽ17b], Appendix A) and hence, admits an abscissa of (absolute) convergence.

Furthermore, still for all $m \in \mathbb{N}^*$ sufficiently large, the abscissa of convergence of $\tilde{\zeta}_{m,\Gamma_{Wm}}^e$ is equal to

$$ D_W = 2 + \frac{\ln \lambda}{\ln b} = 2 - \ln_b \frac{1}{\lambda}. $$

As is proved in [DL23b], $\tilde{\zeta}_{m,\Gamma_{Wm}}^e$, the $m^{th}$ local tube zeta function of the Weierstrass IFD, is the contribution of the $m^{th}$ prefractal approximation $\Gamma_{Wm}$ to $\tilde{\zeta}_{\Gamma_W}^e$, the global effective tube zeta function of the Weierstrass IFD.

*Proof.* Since, by definition (see Definition 4.4, on page 75),

$$ \tilde{\zeta}_{m, \Gamma_{Wm}}^{e}(s) = \int_{0}^{\varepsilon_{m}^{m}} t^{s-3} \tilde{V}_{m, \Gamma_{Wm}}(t) dt, \qquad (\mathbb{R} 71) $$

for all $s \in \mathbb{C}$ with $\mathcal{R}e(s)$ sufficiently large (in fact, for $\mathcal{R}e(s) > D_W$), and according to Theorem 4.8, on page 76 in Section 4.2,

$$ \begin{aligned} \tilde{V}_{m, \Gamma_{Wm}} (\varepsilon_m^m) = & \tilde{V}_{m, \Gamma_{Wm}, \text{Rectangles}} (\varepsilon_m^m) + \tilde{V}_{m, \Gamma_{Wm}, \text{wedges}} (\varepsilon_m^m) \\ & + \tilde{V}_{m, \Gamma_{Wm}, \text{extra outer triangles}} (\varepsilon_m^m) + \tilde{V}_{m, \Gamma_{Wm}, \text{parallelograms}} (\varepsilon_m^m), \end{aligned} \qquad (\mathbb{R} 72) $$
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we have that (still for $\mathcal{R}e(s) > D_W$),

$$ \tilde{\zeta}_{m, \Gamma_{W_m}}^e(s) = \tilde{\zeta}_{m, \text{Rectangles}}^e(s) + \tilde{\zeta}_{m, \text{wedges}}^e(s) + \tilde{\zeta}_{m, \text{extra outer triangles}}^e(s) + \tilde{\zeta}_{m, \text{parallelograms}}^e(s), \quad (\mathcal{R} 73) $$

and still in light of Theorem 4.7, on page 72, combined with the definition of $\tilde{\mathcal{V}}_{m, \Gamma_{W_m}}(t)$, for any $t \in [0, \varepsilon_m^m]$, and of $\tilde{\zeta}_{m, \Gamma_{W_m}}^e = \tilde{\zeta}_{m, \Gamma_{\mathcal{F}_m}}^e$ (see Definition 4.3, on page 62, where $\mathcal{F}_m = \Gamma_{W_m}$ denotes the $m^{th}$ component of the Weierstrass IFD – or, equivalently, the $m^{th}$ prefractal approximation to the Weierstrass Curve), it follows that, for all $m \in \mathbb{N}^*$ sufficiently large, $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ has a meromorphic continuation to all of $\mathbb{C}$ given by formula (R70) in Theorem 4.8, on page 76.

Finally, the fact that, for all $m$ sufficiently large, the abscissa of convergence of $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ coincides with $D_W$ follows by combining formula (R70), on page 76 (for all $s \in \mathbb{C}$) and the method of proof of Theorem 2.1 on page 57 in [LRŽ17b].

Alternatively, the fact that, for all $m \in \mathbb{N}$ sufficiently large, the abscissa of convergence $D_m$ of $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ is equal to

$$ D_W = 2 + \frac{\ln \lambda}{\ln b} = 2 - \ln_b \frac{1}{\lambda}, \quad (\mathcal{R} 74) $$

follows from relation (R70), given on page 76. Indeed, by definition, $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ is a tamed Dirichlet-type integral (DTI), in the sense of [LRŽ17b], Appendix A, Definitions A.1.2 and A.1.3, on page 579. Hence, since $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ is meromorphic in all of $\mathbb{C}$ and, in particular, in a neighborhood of $D_W$, the abscissa of convergence of $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ exists and coincides with the largest real part of the poles of $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$; that is, here, in light of relation (R70) and of Theorem 4.10, on page 81 below (a corollary of the above Theorem 4.8, given on page 76, and which implies that $D_W$ is an actual pole of $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$), $D_m$ coincides with $D_W$, as given by relation (R74) above.

The fact that the first series, $\Sigma_{\text{Rectangles}} = \Sigma_{\text{Rectangles}}(t)$ (appearing in relation (R76)), is locally uniformly convergent (and hence, pointwise convergent), follows from the following uniform estimate (valid for all $s \in \mathbb{C}$, with $\mathcal{R}e(s) \ge \alpha$, where $\alpha \in \mathbb{R}$ is arbitrary),

$$ \begin{aligned} \forall (k, \ell) \in \mathbb{N} \times \mathbb{Z} : \left| (\varepsilon_m^m)^{s-D_W+k(2-D_W)-i\ell} \mathbf{p} \right| &\le (\varepsilon_m^m)^{\alpha-D_W+k(2-D_W)-i\ell} \mathbf{p} \\ &\le \left( \left(\frac{1}{2}\right)^{2-D_W} \right)^k, \quad (\mathcal{R} 75) \end{aligned} $$

provided $0 < \varepsilon_m^m \le \frac{1}{2}$, which is true for all $m$ sufficiently large.

More specifically, we combine the uniform estimate of relation (R75), on page 77, together with the fact that, for $(k, \ell) \in \mathbb{N} \times \mathbb{Z}$ and independently of $m \in \mathbb{N}^*$ large enough), the coefficients $f_{k, \ell, \text{Rectangles}}$ are uniformly bounded.

Also, we reason in exactly the same manner with each of the two double sums in relation (R70), on page 76, defining the remaining effective tube zeta functions contributing to $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$.

It then suffices to apply the same reasoning as the one described in Remark 4.7, on page 80 just below to conclude that, for all $m$ large enough, $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ is meromorphic on all of $\mathbb{C}$, as desired.
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[49] L.O.R. Olsen, Review: fractal geometry and number theory. Complex dimensions of fractal strings and zeros of zeta functions by M. L. Lapidus and M. van Frankenhuijsen, Birkhäuser, Bull. Lond. Math. Soc. 33 (2001) 254–255.
Next, we justify the fact that, for all $s \in \mathbb{C}$, $\tilde{\zeta}_{m, \Gamma_{W_m}}^e(s)$ is given by relation (R70) in Theorem 4.8, on page 76 above.

In order to see this, we apply Definition 4.3, on page 62, of the $m^{th}$ effective tubular volume $\tilde{V}_{m, \Gamma_{W_m}} = \tilde{V}_{m, \Gamma_{W_m}}(t)$, for all $t \in [0, \varepsilon_m^m]$. Accordingly, as was alluded to above, for these values of $t$, and for all $m \in \mathbb{N}^*$ sufficiently large, $\tilde{V}_{m, \Gamma_{W_m}}(t)$ is given by (the sum of) the fractal power series appearing on the right-hand side of relation (R56), on page 72 (or, equivalently, in relation (R57), on page 73), in the fractal tube formula for the Weierstrass IFD obtained in Theorem 4.7, on page 72, but where $\varepsilon_m^m$ is replaced by $t \in [0, \varepsilon_m^m]$.

Then, the same estimate as in relation (R75), on page 77 just above, but now still with $\varepsilon_m^m$ replaced by $t$, and $m_0$ large enough such that $0 < \varepsilon_q^q \le \frac{1}{2}$, for all $q \ge m_0$ (and hence, also, $0 < t \le \frac{1}{2}$) shows that the general term of the first series, namely,

$$ \sum_{k \in \mathbb{N}, \ell \in \mathbb{Z}} \left(\frac{1}{k}\right) \frac{N_b^{1-k(2-D_W)} - 1}{N_b^{1-k(2-D_W)}} \frac{(N_b - 1)^{-i \ell \mathbf{P}}}{(1 - k(2 - D_W)) \ln N_b + 2i\ell\pi} t^{2-D_W+k(2-D_W)-i\ell\mathbf{P}}, $$

appearing in the first term of the right-hand side of relation (R55) in Theorem 4.7, on page 72 (yielding $V_{m, \Gamma_{W_m}}(\varepsilon_m^m)$) implies easily that

$$ \begin{aligned} \Sigma_{\text{Rectangles}}(t) &= \int_0^{\varepsilon_m^m} t^{s-3} \sum_{k \in \mathbb{N}, \ell \in \mathbb{Z}} \left(\frac{1}{k}\right) \frac{N_b^{1-k(2-D_W)} - 1}{N_b^{1-k(2-D_W)}} \frac{(N_b - 1)^{-i \ell \mathbf{P}}}{(1 - k(2 - D_W)) \ln N_b + 2i\ell\pi} t^{2-D_W+k(2-D_W)-i\ell \mathbf{P}} dt \\ &= \sum_{k \in \mathbb{N}, \ell \in \mathbb{Z}} \left(\frac{1}{k}\right) \frac{N_b^{1-k(2-D_W)} - 1}{N_b^{1-k(2-D_W)}} \frac{(N_b - 1)^{-i \ell \mathbf{P}}}{(1 - k(2 - D_W)) \ln N_b + 2i\ell\pi} \int_0^{\varepsilon_m^m} t^{s-3} t^{2-D_W+k(2-D_W)-i\ell \mathbf{P}}, \end{aligned} \quad (R 76) $$

viewed as a function of $t \in [0, \varepsilon_m^m]$, still for a fixed $m \ge m_0$ – converges normally (and thus also, uniformly) in $t$ on $[0, \varepsilon_m^m]$. The same reasoning can be applied to each of the remaining series; i.e.,

$$ \Sigma_{\text{wedges}}^1(t) = \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i \ell \mathbf{P}} t^{3-i \ell \mathbf{P}}}{\ln N_b + 2i \ell \pi}, \quad (R 77) $$

$$ \Sigma_{\text{wedges}}^2(t) = \sum_{k \in \mathbb{N}, \ell \in \mathbb{Z}} \frac{(-1)^k}{2k+1} \frac{N_b^{((2k+1)D_W-2k)} - 1}{N_b^{((2k+1)D_W-2k)}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{((2k+1)D_W-2k)} - 1)^{-i \ell \mathbf{P}} t^{2k+1-i \ell \mathbf{P}}}{((2k+1)D_W-2k) \ln N_b + 2i \ell \pi}, \quad (R 78) $$

$$ \Sigma_{\text{wedges}}^3(t) = \sum_{k \in \mathbb{N}, \ell \in \mathbb{Z}} \frac{(-1)^k}{2k+1} \frac{N_b^{(2k+1)(D_W-1)} - 1}{N_b^{(2k+1)(D_W-1)}} \frac{(N_b^{(2k+1)(D_W-1)} - 1)^{-i \ell \mathbf{P}} t^{5+2k-i \ell \mathbf{P}}}{(2k+1)(D_W-1) \ln N_b + 2i \ell \pi}, \quad (R 79) $$

$$ \Sigma_{\text{triangles and parallelograms}}(t) = \sum_{\ell \in \mathbb{Z}} \frac{(N_b^{2-3D_W} - 1)^{-i \ell \mathbf{P}}}{(2 - 3D_W) \ln N_b + 2i \ell \pi} t^{2-i \ell \mathbf{P}}, \quad (R 80) $$

appearing on the right-hand side of the second equality of relation (R55) in Theorem 4.7, on page 72. Hence, by Weierstrass' theorem (for uniformly convergent series of functions), we can interchange series and integrals in the expression for $\tilde{\zeta}_{m, \Gamma_{W_m}}^e(s)$, given for a fixed arbitrary $s \in \mathbb{C}$, such that $\mathrm{Re}(s) > D_W$, by the truncated Mellin transform,

$$ \tilde{\zeta}_{m, \Gamma_{W_m}}^e(s) = \int_0^{\varepsilon_m^m} t^{s-3} \tilde{V}_{m, \Gamma_{W_m}}(t) dt. \quad (R 81) $$

In fact, with the notation of Properties 4.3–4.6, on pages 65–71, we have that (still for all $m \ge m_0$, $\tilde{\zeta}_{m, \Gamma_{W_m}}^e(s)$ is given by relation (R57), on page 73, first for all $s \in \mathbb{C}$ with $\mathrm{Re}(s) \ge D_W$ – and then, by
[50] L.O.R. Olsen, Multifractal tubes, in: Further Developments in Fractals and Related Fields, in: Trends Math., Birkhäuser/Springer, New York, 2013, pp. 161–191.

[51] L.O.R. Olsen, Multifractal tubes: multifractal zeta-functions, multifractal Steiner formulas and explicit formulas, in: D. Carfi, M.L. Lapidus, E.P.J. Pearse, M. van Frankenhuijsen (Eds.), Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. I. Fractals in Pure Mathematics, in: Contemp. Math., vol. 600, Amer. Math. Soc., Providence, RI, 2013, pp. 291–326.

[52] F. Przytycki, M. Urbański, On the Hausdorff dimension of some fractal sets, Stud. Math. 93 (2) (1989) 155–186.

[53] W. Shen, Hausdorff dimension of the graphs of the classical Weierstrass functions, Math. Z. 289 (2018) 223–266.

[54] E.C. Titschmarsh, The Theory of Functions, second edition, Oxford University Press, 1939.

[55] K. Weierstrass, Über continuirliche Funktionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten Differential quotienten besitzen, J. Reine Angew. Math. 79 (1875) 29–31.

[56] A. Zygmund, Trigonometric Series. Vols. I, II, third edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002, with a foreword by Robert A. Fefferman.
the principle of analytic (i.e., meromorphic) continuation, for all $s \in \mathbb{C}$, since, as was explained above, each of the series in relations $(\mathbb{R}76)-(\mathbb{R}80)$, on pages 78–78 above, converges and is a meromorphic function of $s$ on all of $\mathbb{C}$.

Here is a direct way to establish the meromorphicity of $\tilde{\zeta}_{m,\Gamma_{W_m}}^e$ (for all $m \ge m_0$) and to identify its (possible) poles, without using the chordal metric on the Riemann sphere (see Remark 4.7, on page 80 below, for a closely related use of this latter metric.)

Let $\omega$ be a potential pole (i.e., a possible Complex Dimension, as given by Theorem 4.10, on page 81 below), say,

$$ \omega = \omega_{k,l} = D_W - k (2 - D_W) + i \ell \mathbf{p}, $$

with $k \in \mathbb{N}$ and $\ell \in \mathbb{Z}$.

Then, by excizing an arbitrary small compact disk $D_\omega$ centered at $\omega$ from a slightly larger open disk $D_\omega^+$ (also centered at $\omega$), it follows, much as in the above discussion, that the corresponding double series of holomorphoc functions in the resulting domain $D_\omega^+ \setminus \overline{D_\omega}$ is normally – and hence, also uniformly – convergent in $D_\omega^+ \setminus \overline{D_\omega}$.

Hence, since $D_\omega$ can be chosen arbitrarily small, we deduce from Weierstrass' theorem for series of holomorphic functions that the sum of the double series appearing in the right-hand side of formula $(\mathbb{R}70)$ in Theorem 4.8, on page 76, is holomorphic in $D_\omega^+ \setminus \overline{D_\omega}$, which is an arbitrary small pointed neighborhood of $\omega$ – and thus, that

$$ \Sigma(s) = \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}} f_{k, \ell, \text{Rectangles}} \frac{(\varepsilon_m^m)^{s-D_W+k(2-D_W)-i\ell \mathbf{p}}}{s - D_W + k(2 - D_W) - i \ell \mathbf{p}} \\ + \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}} \left\{ f_{\ell, k, \text{wedges}, 1} \frac{(\varepsilon_m^m)^{s+1-i\ell \mathbf{p}}}{s+1-i\ell \mathbf{p}} + f_{\ell, k, \text{wedges}, 2} \frac{(\varepsilon_m^m)^{s+2k-1-i\ell \mathbf{p}}}{s+2k-1-i\ell \mathbf{p}} + f_{\ell, k, \text{wedges}, 3} \frac{(\varepsilon_m^m)^{s+3+2k-i\ell \mathbf{p}}}{s+3+2k-i\ell \mathbf{p}} \right\} \\ + \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}} f_{k, \ell, \text{triangles, parallelograms}} \frac{(\varepsilon_m^m)^{s-1-i\ell \mathbf{p}}}{s-1-i\ell \mathbf{p}} \quad (\mathbb{R} 82) $$

is holomorphic away from any potential singularity $\omega_{k,\ell} = D_W - k (2 - D_W) + i \ell \mathbf{p}$.

Now, by using the uniform convergence in $D_\omega^+ \setminus \overline{D_\omega}$, we can interchange limits and deduce that the following limits exist in $\mathbb{C}$, and are given as follows:

$$ \mathrm{res} (\Sigma, \omega_{k,l}) = \lim_{s \to \omega_{k,l}} (s - \omega_{k,l}) \Sigma(s), \quad (\mathbb{R} 83) $$

from which we deduce that $\Sigma$ has at most a simple pole at $\omega = \omega_{k,l}$. Since $f_{k,l,\text{Rectangles}} \ne 0$, then $\omega = \omega_{k,l}$ is a simple pole of $\Sigma$, with associated residue $f_{k,l} \ne 0$, as implied by formula (R83).

We conclude from the above discussion that $\Sigma$ is meromorphic in all of $\mathbb{C}$, with potential poles (necessarily simple poles) the possible Complex Dimensions listed in Theorem 4.10, on page 81 below. Since we know that still for all sufficiently large values of the positive integer $m$,

$$ \tilde{\zeta}_{m, \Gamma_{W_m}}^e(s) = \Sigma(s), \quad (\mathbb{R} 84) $$

for all $s$ in the domain (open right-half plane) $\mathbb{R}e(s) > D_W$, we deduce from the principle of analytic (i.e., meromorphic) continuation that $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ has a meromorphic continuation to all of $\mathbb{C}$, coinciding
with $\Sigma$ in $\mathbb{C}$ – and hence, having the same potential (as well as actual) poles as $\Sigma$, and the same associated residues.

This completes the proof of Theorem 4.8 (page 76), which will also be used in part in order to prove Theorem 4.10, on page 81 (about the possible Complex Dimensions of $\tilde{\zeta}_{m,\Gamma_{W_m}}$) and Remark 4.7, on page 80 below. $\square$

**Remark 4.7.** The fact that the global effective tube zeta function $\tilde{\zeta}_{\Gamma_W}^e$ admits a meromorphic continuation to all of $\mathbb{C}$ is obtained by applying Weierstrass' theorem for (locally) uniformly convergent sequences of holomorphic functions. First, we note that, for all sufficiently large $m \in \mathbb{N}$, the set $\mathcal{Z}$ of possible poles of the local tube zeta function $\tilde{\zeta}_{m,\Gamma_{W_m}}^e(s)$ does not depend on $m$, and is given by Theorem 4.10, pag 81 below. Note that $\mathcal{Z}$ is discrete, and thus closed in $\mathbb{C}$. It then makes sense to consider any of those poles, that we will denote by $\omega$. The local tube zeta function $\tilde{\zeta}_{m,\Gamma_{W_m}}^e$ is then holomorphic on the connected open subset of $\mathbb{C}$ given by $\mathbb{C} \setminus \mathcal{Z}$. We can clearly see that the sequence of functions $(\tilde{\zeta}_{m,\Gamma_{W_m}}^e)_{m \ge m_0}$ converges normally (and hence, uniformly) in a connected open (and relatively compact) neighborhood of any given $\omega \in \mathcal{Z}$ – i.e., for $s = x + iy \in \mathbb{C}$ close to $\omega$. Weierstrass' theorem, applied once again, then ensures the holomorphicity of the limit $\tilde{\zeta}_{\Gamma_W}^e$ on the domain $\mathbb{C} \setminus \mathcal{Z}$. It follows that the global tube zeta function $\tilde{\zeta}_{\Gamma_W}^e$ is meromorphic in all of $\mathbb{C}$, with possible set of poles given by $\mathcal{Z}$.

**Corollary 4.9 (of Theorem 4.8, on page 76) Local and Global Distance Zeta Function for the Weierstrass Iterated Fractal Drums).**

By analogy with the functional equation given in [LRŽ17b] (Theorem 2.2.1, page 112), along with Theorem 4.8, on page 76 just above, given the cohomology infinitesimal $\epsilon$ (see Definition 3.1, on page 37), the global effective distance zeta function $\zeta_{\Gamma_W}^e$ is given, for any complex number $s$, by the following expression:

$$ \zeta_{\Gamma_W}^e(s) = \lim_{m \to \infty} \zeta_{m, \Gamma_{W_m}}^e(s), \qquad (\mathbb{R} 85) $$

where, for all $m \in \mathbb{N}$ sufficiently large, $\zeta_{m,\Gamma_{W_m}}^e$, the $m^{th}$ local effective distance zeta function of the Weierstrass IFD, is given, for any complex number $s$, by

$$ \begin{aligned} \zeta_{m, \Gamma_{W_m}}^e(s) &= (\varepsilon_m^m)^{s-2} \tilde{V}_{m, \Gamma_{W_m}}(\varepsilon_m^m) + (2-s) \int_0^{\varepsilon_m^m} t^{s-3} \tilde{V}_{m, \Gamma_{W_m}}(t) dt \\ &= (\varepsilon_m^m)^{s-2} \tilde{V}_{m, \Gamma_{W_m}}(\varepsilon_m^m) + (2-s) \zeta_{m, \Gamma_{W_m}}^e(s), \end{aligned} \qquad (\mathbb{R} 86) $$

where $\tilde{V}_{m,\Gamma_{W_m}}$ denotes the $m^{th}$ local effective tubular volume obtained in relations (R56)-(R57) of Theorem 4.7, on page 72, and where $\tilde{\zeta}_{m,\Gamma_{W_m}}^e(s)$ is given in relation (R70) of Theorem 4.8, on page 76 (note that, by construction, $\tilde{V}_{m,\Gamma_{W_m}}(\varepsilon_m^m) = V_{m,\Gamma_{W_m}}(\varepsilon_m^m)$). The first equality in relation (R86) is only valid for

$$ \mathrm{Re}(s) > D_m = D_W, $$

while the last one is valid for all $s$ in $\mathbb{C}$. Furthermore, still for all $m \in \mathbb{N}$ sufficiently large, the distance zeta function $\zeta_{m,\Gamma_{W_m}}^e$ admits a meromorphic continuation to all of $\mathbb{C}$, given by the last equality of relation (R86) just above, with $\tilde{\zeta}_{m,\Gamma_{W_m}}^e$ given as in Theorem 4.8, on page 76.
Remark 4.8. It follows from the above functional equation (R86), on page 80, as well from the general theory developed in [LRŽ17b], that $\zeta_{m,\Gamma_{W_m}}^e$ and $\tilde{\zeta}_{m,\Gamma_{W_m}}^e$ have exactly the same poles, with precisely related residues, for simple poles, which is the case here. Hence, they define the same Complex Dimensions. In light of Remark 4.7, on page 80 above, an analogous comment can be made about the global effective tube and distance zeta functions $\tilde{\zeta}_{\Gamma_W}^e$ and $\zeta_{\Gamma_W}^e$.

We recall from [LRŽ17b] that the Complex Dimensions are defined as the poles of the meromorphic continuation of the tube (or, equivalently, the distance) zeta function. In our present setting, the set of Complex Dimensions of the Weierstrass IFD is the set of Complex Dimensions of the sequence of Weierstrass IFDs introduced in Remark 3.3, on page 60. Hence, those Complex Dimensions are the poles of the effective tube zeta functions – or, equivalently, the effective distance zeta functions – associated to those IFDs, respectively obtained in Theorem 4.8, on page 76 and Corollary 4.9, page 80 above.

Remarkably, in light of Theorem 4.8, on page 76, it turns out that the set of (possible) Complex Dimensions, defined as the set of (possible) poles of the $m^{th}$ local effective tube zeta function $\zeta_{m,\Gamma_{W_m}}^e$ (or, equivalently, of $\zeta_{m,\Gamma_{W_m}}^e$), does not change, for all sufficiently large $m \in \mathbb{N}^*$; i.e., this set of (possible) Complex Dimensions – viewed as a multiset taking into account the multiplicities of the possible poles – stabilizes for all sufficiently large $m \in \mathbb{N}^*$.

By definition, this set is then called the set of (possible) Complex Dimensions of the Weierstrass IFD $\Gamma_W^I$.

We expect this “stabilization phenomenon” to be common to a large class of IFDs associated with complicated fractals.

Observe that also in light of Theorem 4.8, on page 76, we could equivalently define the set of (possible) Complex Dimensions of the Weierstrass IFD as the set of (possible) poles of the global effective tube zeta function $\tilde{\zeta}_{\Gamma_W}^e$ (or, equivalently, of the global effective distance zeta function $\zeta_{\Gamma_W}^e$) of the Weierstrass IFD.

**Theorem 4.10 (Complex Dimensions of the Weierstrass IFD).**

The possible Complex Dimensions of the Weierstrass IFD $\Gamma_W^I$ are all simple, and given as follows:

$$D_W - k(2 - D_W) + i\ell \mathbf{p}, \quad \text{with } k \in \mathbb{N}, \ell \in \mathbb{Z},$$

$$1 - 2k + i\ell \mathbf{p}, \quad \text{with } k \in \mathbb{N}, \ell \in \mathbb{Z}, \text{ along with } -2 \text{ and } 0,$$

where $\mathbf{p} = \frac{2\pi}{\ln N_b}$ is the oscillatory period of the Weierstrass IFD.

Furthermore, the one-periodic functions (with respect to the variable $\ln N_b$ $(\varepsilon_m^m)^{-1}$, see Property 4.1, on page 61), respectively associated to the values $D_W - k(2 - D_W)$, $k \in \mathbb{N}$, are nonconstant. (See also Subsection 4.3.2, on page 83 below for the exceptional cases.)

In addition, all of the Fourier coefficients of the latter periodic functions are nonzero, which implies that there are infinitely many Complex Dimensions that are nonreal, including all of those with maximal real part $D_W$, which are the principal Complex Dimensions, in the terminology of [LRŽ17b], and therefore give rise to geometric oscillations (or vibrations) with the largest amplitude, in the fractal tube formula obtained in Theorem 4.7, on page 72 above and reformulated in Theorem 4.11, on
Finally, for each $k \in \mathbb{N}$ and $\ell \in \mathbb{Z}$, $D_W - k(2 - D_W) + i\ell\mathbf{p}$, $1 + i\ell\mathbf{p}$, $-2$ and $0$ are all simple Complex Dimensions of the Weierstrass IFD; i.e., they are simple poles of the $m^{th}$ tube (or, equivalently, of the distance) zeta functions, for all $m \in \mathbb{N}^*$ sufficiently large.

Consequently, the Weierstrass IFD $\Gamma_W^\ell$ is fractal, in the sense of the theory of Complex Dimensions developed in [LvF00], [LvF06], [LvF13], [LRŽ17b] and [Lap19].

We refer to Subsection 4.3.2, on page 83, for a discussion of the exceptional cases, and to Subsection 4.3.3, on page 85 for a possible interpretation of our results.

*Proof.* The proof of this theorem is included in the latter part of the proof of Theorem 4.8, on page 76. $\square$

*Remark 4.9.* The justification of this remark is also included in the latter part of proof of Theorem 4.8, given on page 76. Note, however, that we are giving here more precise statements and informations than in the aforementioned proof.

i. Let $m \in \mathbb{N}$ be arbitrary, but sufficiently large, so that both Theorem 4.8 (page 76) and Corollary 4.9 (page 80) are valid. Let $\omega$ be a potential pole (necessary simple) of $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ – or, equivalently, of $\zeta_{m, \Gamma_{W_m}}^e$ (since $D_W < 2$); $\omega$ is a possible Complex Dimension of the Weierstrass IFD, as given in Theorem 4.10, on page 81.

Say, for notational simplicity, that

$$ \omega = \omega_{k,\ell} = D_W - k(2 - D_W) + i\ell\mathbf{p}, \qquad (\mathbb{R} 87) $$

for some $k \in \mathbb{N}$ and $\ell \in \mathbb{Z}$. Then, with the notation and the latter part of Theorem 4.8, given on page 76, we have that

$$ \mathrm{res} \left( \tilde{\zeta}_{m, \Gamma_{W_m}}^e, \omega_{k, \ell} \right) = \lim_{s \to \omega_{k, \ell}} (s - \omega_{k, \ell}) \tilde{\zeta}_{m, \Gamma_{W_m}}^e(s) = f_{k, \ell, \mathrm{Rectangles}} \qquad (\mathbb{R} 88) $$

and

$$ \mathrm{res} \left( \zeta_{m, \Gamma_{W_m}}^e, \omega_{k, \ell} \right) = \lim_{s \to \omega_{k, \ell}} (s - \omega_{k, \ell}) \zeta_{m, \Gamma_{W_m}}^e(s) = (2 - \omega_{k, \ell}) f_{k, \ell, \mathrm{Rectangles}} = (2 - \omega_{k, \ell}) \mathrm{res} \left( \tilde{\zeta}_{m, \Gamma_{W_m}}^e, \omega_{k, \ell} \right), \qquad (\mathbb{R} 89) $$

where the last equality follows from the functional equation connecting $\zeta_{m, \Gamma_{W_m}}^e$ and $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ (much as in [LRŽ17b]), and as stated in relation (R86) in Corollary 4.9, on page 80. Therefore, we see (much as in the end of the proof of Theorem 4.7, page 72), that $\omega = \omega_{k, \ell}$ is a pole (necessarily a simple pole of $\zeta_{m, \Gamma_{W_m}}^e$, or, equivalently, of $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$) – i.e., $\omega$ is a simple Complex Dimension of the Weierstrass IFD – if and only if $f_{k, \ell, \mathrm{Rectangles}} \neq 0$, which, according to Theorem 4.7, on page 72, is always the case.

Furthermore, in this case, the residue of $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ (respectively, $\zeta_{m, \Gamma_{W_m}}^e$) at $\omega$ is given by relation (R88) (resp., by relation (R89) just above.
ii. Moreover, also in agreement with the higher-dimensional theory developed in [LRŽ17b] (see also [LRŽ17a] and [LRŽ18], for example), the Complex Dimensions of the Weierstrass IFD can be defined indifferently via the $m^{th}$ local effective tube zeta functions $\tilde{\zeta}_{m,\Gamma_{W_m}}^e$ or via the $m^{th}$ local effective distance zeta functions $\zeta_{m,\Gamma_{W_m}}^e$, for all $m \in \mathbb{N}^*$ sufficiently large.

iii. Parts i. and ii. of this remark are valid both for the potential (or possible) Complex Dimensions and for the exact Complex Dimensions of the Weierstrass IFD.

**Theorem 4.11 (Condensed Fractal Tube Formula for The Weierstrass IFD (Corollary of Theorem 4.7, on page 72).**

Given $m \in \mathbb{N}$ sufficiently large, the tubular effective volume $\tilde{\mathcal{V}}_{m,\Gamma_{W_m}}(\varepsilon_m^m)$ of the $\varepsilon_m^m$-neighborhood $\mathcal{D}(\varepsilon_m^m)$ of the Weierstrass IFD, can be expressed in the following manner:

$$ \tilde{\mathcal{V}}_{m,\Gamma_{W_m}}(\varepsilon_m^m) = \sum_{k=0}^{\infty} (\varepsilon_m^m)^{2-(D_W-k(2-D_W))} G_{k,D_W} \left( \ln_{N_b}\left(\frac{1}{\varepsilon_m^m}\right) \right) + \sum_{k=0}^{\infty} (\varepsilon_m^m)^{2-(1-2k)} G_{k,1} \left( \ln_{N_b}\left(\frac{1}{\varepsilon_m^m}\right) \right) + \pi (\varepsilon_m^m)^2 - \frac{\pi (\varepsilon_m^m)^4}{2}, \quad (\mathcal{R} 90) $$

where, for any fixed (but arbitrary) $k \in \mathbb{N}$, $G_{k,D_W}$ and $G_{k,1}$ denote, respectively, continuous one-periodic functions (with respect to the variable $\ln_{N_b}(\varepsilon_m^m)^{-1}$, see Property 4.1, on page 61) associated to all of the Complex Dimensions of real parts $D_W - k(2 - D_W)$ and $1 - 2k$. Furthermore, all of the Fourier coefficients of the periodic functions $G_{k,D_W}$ (for any $k \in \mathbb{N}$) and $G_{0,1}$ are nonzero. In particular, these periodic functions are not constant. Moreover, the functions $G_{0,D_W}$ and $G_{0,1}$ are bounded away from zero and infinity.

This amounts to an expression of the form

$$ \tilde{\mathcal{V}}_{m,\Gamma_{W_m}}(\varepsilon_m^m) = \sum_{\substack{\alpha \text{ real part of a Complex Dimension} \\ \alpha \notin \{-2,0\}}} (\varepsilon_m^m)^{2-\alpha} G_{\alpha} \left( \ln_{N_b} \left( \frac{1}{\varepsilon_m^m} \right) \right) + \pi (\varepsilon_m^m)^2 - \frac{\pi (\varepsilon_m^m)^4}{2}, \quad (\mathcal{R} 91) $$

where, for any real part $\alpha$ of a Complex Dimension, with $\alpha \notin \{-2,0\}$, $G_{\alpha}$ denotes a continuous and one-periodic function.

### 4.3.2 Exceptional Cases

One might naturally question the following exceptional cases:

i. $D_W - k_0(2 - D_W) = 0$, for some $k_0 \in \mathbb{N}$, which occurs when

$$ D_W = \frac{2k_0}{1+k_0}, \text{ i.e., } 2 + \frac{\ln \lambda}{\ln N_b} = \frac{2k_0}{1+k_0}, \text{ or } \lambda = N_b^{-\frac{2}{1+k_0}}. $$

According to the terminology of [LRŽ17b], Chapter 4, or [LvF06], Chapter 12, this first case corresponds to the situation when the Weierstrass Curve is fractal in dimension 0. We then happen to have a discrete line of Complex Dimensions with real part 0,
$$ \mathcal{L}_0 = \{0 + i \ell \mathbf{p}, \ell \in \mathbb{Z}\} = \{i \ell \mathbf{p}, \ell \in \mathbb{Z}\}, $$

which is obtained by merger with the discrete line of actual Complex Dimensions,

$$ \mathcal{L}_{D_W, k_0} = \left\{ \underbrace{D_W - k_0 (2 - D_W)}_{0 \text{ here}} + i \ell \mathbf{p}, \ell \in \mathbb{Z} \right\}. $$

Note that the *actual* Complex Dimensions are *not double* (i.e., of multiplicity two). This directly comes from the expression obtained in relation (R70) of Theorem 4.8, on page 76 for the effective fractal tube zeta function $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$, which becomes here, for all $m$ sufficiently large, and for any complex number $s$,

$$ \begin{aligned} \tilde{\zeta}_{m, \Gamma_{W_m}}^e(s) &= \sum_{\ell \in \mathbb{Z}} f_{k, \ell, \text{Rectangles}} \frac{(\varepsilon_m^m)^{s-i\ell\mathbf{p}}}{s - i\ell\mathbf{p}} \\ &= \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}, k \neq k_0} f_{k, \ell, \text{Rectangles}} \frac{(\varepsilon_m^m)^{s-D_W+k(2-D_W)-i\ell\mathbf{p}}}{s - D_W + k(2-D_W) - i\ell\mathbf{p}} \\ &\quad + \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}} \left( f_{k, \ell, \text{wedges}, 1} \frac{(\varepsilon_m^m)^{s+1-i\ell\mathbf{p}}}{s+1-i\ell\mathbf{p}} + f_{k, \ell, \text{wedges}, 2} \frac{(\varepsilon_m^m)^{s+2k-1-i\ell\mathbf{p}}}{s+2k-1-i\ell\mathbf{p}} + f_{\ell, k, \text{wedges}, 3} \frac{(\varepsilon_m^m)^{s+3+2k-i\ell\mathbf{p}}}{s+3+2k-i\ell\mathbf{p}} \right) \\ &\quad + \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}} f_{\ell, k, \text{triangles, parallelograms}} \frac{(\varepsilon_m^m)^{s-1-i\ell\mathbf{p}}}{s-1-i\ell\mathbf{p}} + \frac{\pi (\varepsilon_m^m)^s}{s} - \frac{\pi (\varepsilon_m^m)^{s+2}}{4(s+2)}, \end{aligned} \quad (R 92) $$

where, as was already seen in Theorem 4.7, on page 72 the notation $f_{k,\ell,\text{Rectangles}}$, $f_{k,\ell,\text{wedges},\ell}$, with $1 \le \ell \le 3$, and $f_{k,\ell,\text{triangles, parallelograms}}$, respectively account for the coefficients associated to the sums corresponding to the contribution of the rectangles, wedges, triangles and parallelograms.

This could also be deduced from the fact if the pole $s=0$ were double, we would have terms involving $\ln \varepsilon_m^m$ in the expression of $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$, because, for any integer $\ell \in \mathbb{Z}$ and any complex number $s$,

$$ (\varepsilon_m^m)^{s-i\ell\mathbf{p}} = e^{(s-i\ell\mathbf{p}) \ln \varepsilon_m^m}; $$

see [LvF06], Subsection 6.1.1, pages 180–182.

The novelty of this case is that we have Complex Dimensions above 0.

ii. $D_W - k_1 (2 - D_W) = 1$, for some $k_1 \in \mathbb{N}$, which occurs when

$$ D_W = \frac{1 + 2k_1}{1 + k_1}; \quad \text{i.e.,} \quad 2 + \frac{\ln \lambda}{\ln N_b} = \frac{1 + 2k_1}{1 + k_1} \quad \text{or, equivalently,} \quad \lambda = N_b^{-\frac{1}{1+k_1}}. $$

Since, here, $\lambda N_b \neq 1$, it follows that $k_1 \neq 0$.

According to the terminology mentioned in i., this second case corresponds to the situation when the Weierstrass Curve is *fractal in dimension 1*. We then happen to have a discrete line of Complex Dimensions with real part 1,
$$ \mathcal{L}_1 = \{1 + i \ell \mathbf{p}, \ell \in \mathbb{Z}\}, $$

which is obtained by merger with the discrete line of actual Complex Dimensions,

$$ \mathcal{L}_{D_W, k_1} = \left\{ \underbrace{D_W - k_1 (2 - D_W)}_{1 \text{ here}} + i \ell \mathbf{p}, \ell \in \mathbb{Z} \right\}. $$

Note again that the *actual* Complex Dimensions are *not double*. As above, this directly comes from the expression obtained in relation (R70) of Theorem 4.8, on page 76 for the fractal tube zeta function $\zeta_{m, \Gamma_{W_m}}^e$, which becomes here, for any complex number $s$,

$$
\begin{align*}
\zeta_{m, \Gamma_{W_m}}^e(s) &= \sum_{\ell \in \mathbb{Z}} (f_{k, \ell_1, \text{Rectangles}} + f_{\ell, 0, \text{wedges}, 2}) \frac{(\varepsilon_m^m)^{s-1-i\ell \mathbf{p}}}{s-1-i\ell \mathbf{p}} \\
&= \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}, k \neq k_1} f_{k, \ell, \text{Rectangles}} \frac{(\varepsilon_m^m)^{s-D_W+k(2-D_W)-i\ell \mathbf{p}}}{s-D_W+k(2-D_W)-i\ell \mathbf{p}} \\
&\quad + \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}^*} f_{k, \ell, \text{wedges}, 2} \frac{(\varepsilon_m^m)^{s+2k-1-i\ell \mathbf{p}}}{s+2k-1-i\ell \mathbf{p}} \\
&\quad + \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}} \left( f_{k, \ell, \text{wedges}, 1} \frac{(\varepsilon_m^m)^{s+1-i\ell \mathbf{p}}}{s+1-i\ell \mathbf{p}} + f_{\ell, k, \text{wedges}, 3} \frac{(\varepsilon_m^m)^{s+3+2k-i\ell \mathbf{p}}}{s+3+2k-i\ell \mathbf{p}} \right) \\
&\quad + \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}} f_{k, \ell, \text{triangles, parallelograms}} \frac{(\varepsilon_m^m)^{s-1-i\ell \mathbf{p}}}{s-1-i\ell \mathbf{p}} + \frac{\pi (\varepsilon_m^m)^s}{s} - \frac{\pi (\varepsilon_m^m)^{s+2}}{4(s+2)}. \tag{R 93}
\end{align*}
$$

What is new in this case is that we are sure that every possible Complex Dimension on $\mathcal{L}_1$, i.e., every complex number $1 + i \ell \mathbf{p}$, with $\ell \in \mathbb{Z}$, is an *actual* Complex Dimension of the Weierstrass Curve, because the same is true for each point of $\mathcal{L}_{D_W, k_1}$.

### 4.3.3 Possible Interpretation

Figure 18, on page 86, gives the distribution of Complex Dimensions. In order to understand their deeper meaning, one may consider an horizontal $\ell \mathbf{p}$ line, of equation $y = \ell \mathbf{p}$, where $\ell \in \mathbb{Z}$ is arbitrary (but fixed). Such a line corresponds to the $\ell^{th}$-order vibration mode, but which can also be interpreted as coming from:

i. The vertical line $x = 0$, or, in other words, oscillations coming from points: indeed, the prefractal graph $\Gamma_{W_m}$ is, at first, constituted of points.

ii. The vertical line $x = 1$, which this time correspond to oscillations coming from lines (or, rather, line segments): prefractal as it is, $\Gamma_{W_m}$ is constituted of lines, in an Euclidean space of dimension two.
<figure><img src="image_20.png" /><figcaption>Figure 18: The Complex Dimensions of the Weierstrass IFD. The nonzero Complex Dimensions are periodically distributed (with the same period $p = \frac{2\pi}{\ln N_b}$, the oscillatory period of the Weierstrass IFD) along countably many vertical lines, with abscissae $D_W - k (2 - D_W)$ and $1 - 2k$, where $k \in \mathbb{N}$ is arbitrary. In addition, 0 and $-2$ are Complex Dimensions of the Weierstrass IFD.</figcaption></figure>

For the sake of representation, there is a different color for each vertical line, and a specific symbol is used to plot the imaginary parts of the Complex Dimensions associated with a given vertical line. (See also Subsection 4.3.2, on page 83 for the exceptional cases.)

iii. The vertical line $x = D_W$, which, this time, corresponds to oscillations coming from the whole prefractal $\Gamma_{W_m}$ itself.

iv. The vertical lines $x = D_W - k (2 - D_W)$, with $k$ in $\mathbb{N}^* = \mathbb{N} \setminus \{0\}$.

For $k \le m$, it corresponds to oscillations coming from the prefractal graphs $\Gamma_{W_{m-k}}$, a phenomenon which can be understood via the following consideration:

Switching from the $(m-k)^{th}$ prefractal graph, to the $m^{th}$ one, $0 < k \le m$, is done by applying $k$ iterates of the $T_j$ maps,

$$T_{j_1 \cdots j_k} = T_{j_1} \circ \cdots \circ T_{j_k} \qquad (\mathcal{R} 94)$$

In terms of the vertical distance between consecutive vertices, this amounts to a multiplication
of the amplitudes by the factor $\lambda^k = N_b^{-k(2-D_W)}$, associated to a sum of cosine expressions.

It thus provides an interesting interpretation of the real parts

$$D_W - k(2 - D_W), \quad \text{for } 0 < k \le m, \qquad (\mathcal{R} 95)$$

insofar as the $m^{th}$ prefractal graph bears – or, in a sense, *feels* – the oscillations of its predecessors.

There remains the lines $x = D_W - k(2 - D_W)$, with $k > m$.

In order to interpret them, one could think in the same way, but, without associated graphs, how? Except if they could exist, in some way. This will be the purpose of a later extension of the prefractal sequence $(\Gamma_{W_m})_{m \in \mathbb{N}}$, a priori indexed by nonnegative integers, to negative ones, via the new concept of *antefractals*. However, this point will not be discussed in the present paper.

### 4.3.4 Analogy with the General Theory of Complex Dimensions

Our results in Theorem 4.7, on page 72 and Theorem 4.11, on page 83 above, on the fractal tube formula for the Weierstrass IFD are similar to the general (exact, pointwise) fractal tube formulas (via either tube or distance zeta functions) obtained in the higher-dimensional theory of Complex Dimensions in [LRŽ17b] (Chapter 5), or in [LRŽ18], and extending the fractal tube formulas for fractal strings obtained in [LvF00] and [LvF06] (Chapter 8). Compare, e.g., in the case of simple poles and under the hypothesis of strong languidity (a strong form of polynomial growth condition) of either $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ or $\zeta_{m, \Gamma_{W_m}}^e$ [LRŽ17b], Theorem 5.1.16, page 427, or Theorem 5.3.17, page 449, respectively.

There is a notable difference, however, due to the great complexity of the Weierstrass Curve $\Gamma_W$ and of the associated IFD $\Gamma_W^I$. Namely, the fractal tube formula is only given for the volume $V_{m, \Gamma_{W_m}}(\varepsilon_m^m)$ of the $m^{th}$ prefractal approximation $\Gamma_{W_m}$, and evaluated at the $m^{th}$ cohomology infinitesimal $\varepsilon_m^m$, for all sufficiently large $m \in \mathbb{N}$.

Indeed, according to the aforementioned results from [LRŽ17b] and [LRŽ18], we would have, in particular, that the tubular volume is given as follows:

$$V_{m, \Gamma_{W_m}} (\varepsilon_m^m) = \sum_{\omega} \mathrm{res} (\tilde{\zeta}_{m, \Gamma_{W_m}}^e, \omega) (\varepsilon_m^m)^{2-\omega} = \sum_{\omega} \frac{\mathrm{res} (\zeta_{m, \Gamma_{W_m}}^e, \omega)}{2-\omega} (\varepsilon_m^m)^{2-\omega}, \qquad (\mathcal{R} 96)$$

where, in each of these two sums, $\omega$ ranges through all of the Complex Dimensions of $\Gamma_W^I$ (i.e., the poles of either $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ or, equivalently, $\zeta_{m, \Gamma_{W_m}}^e$).

Recall from equation (R40)–(♦♦) in Remark 4.3, on page 65 above that

$$\mathrm{res} (\zeta_{m, \Gamma_{W_m}}^e, \omega) = (2 - \omega) \mathrm{res} (\tilde{\zeta}_{m, \Gamma_{W_m}}^e, \omega). \qquad (\mathcal{R} 97)$$

In order to obtain the fractal tube formula in Theorem 4.7, on page 72 (and hence also, in Theorem 4.11, on page 83), however, we did not need to appeal to the aforementioned results of the general theory, by first calculating $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ or $\zeta_{m, \Gamma_{W_m}}^e$ (using their basic scaling and symmetry properties described in [LRŽ17b], along with the geometric properties of $\Gamma_W$ described in Section 2 above) and then, verifying that the appropriate notion of strong languidity is satisfied. This could have been
done, but was unnecessary in our present situation.

Instead, as was explained earlier, we first directly calculated the tubular volume $V_{m,\Gamma_{W_m}} (\varepsilon_m^m)$ in Theorem 4.7, on page 72, and then deduced from the resulting fractal tube formula, via Mellin transformation, an explicit expression for the $m^{th}$ local effective tube zeta function $\tilde{\zeta}_{m,\Gamma_{W_m}}^e$ – and further, for the $m^{th}$ local effective distance zeta function $\zeta_{m,\Gamma_{W_m}}^e$, via the functional equation recalled in relation (♦) of Remark 4.3, on page 65. Finally, as would have been the case if we had adopted the first method outlined above, we deduced (in Theorem 4.10, on page 81) the values of the (possible) Complex Dimensions of the Weierstrass IFD $\Gamma_W^I$, as the poles of $\tilde{\zeta}_{m,\Gamma_{W_m}}^e$ (or, equivalently, of $\zeta_{m,\Gamma_{W_m}}^e$, since $D_W < 2$).

## 4.4 Minkowski Dimension, Minkowski Nondegeneracy, and Average Minkowski Content

We next obtain new and refined results concerning the geometry – and, in particular, the Minkowski nondegeneracy, non Minkowski measurability, as well as the average Minkowski content of the Weierstrass IFD. For this purpose, and for the benefit of the reader who may not be familiar with these notions, we first state several definitions, which are now suitably adapted to our current setting and to the notions of effective tubular volumes.

In the spirit of the remainder of this paper, the definition of (upper, lower) Minkowski contents and dimensions, for example, will be given in terms of the cohomology infinitesimal $(\varepsilon_m^m)_{m=0}^\infty$, viewed as a sequence of positive scales tending to zero, as $m \to \infty$. So will the notions of Minkowski nondegeneracy and Minkowski measurability, as well as of effective average Minkowski content.

**Definition 4.5 (Lower and Upper r-Dimensional Minkowski Contents – Lower and Upper Minkowski Dimensions, and Minkowski Dimension of an IFD).**

Let $\mathcal{F}^I$ be an arbitrary iterated fractal drum of $\mathbb{R}^2$; see Definition 3.2, on page 38. More precisely, we hereafter consider the sequence of ordered pairs $(\mathcal{F}_m, \varepsilon_{\mathcal{F},m})_{m \in \mathbb{N}}$, where, for each $m \in \mathbb{N}$, $\mathcal{F}_m$ is the $m^{th}$ prefractal approximation to a fractal set $\mathcal{F}$, and where $\varepsilon_{\mathcal{F},m}^m$ is the associated $m^{th}$ cohomology infinitesimal.

Then, given $r \ge 0$, $m \in \mathbb{N}$, and the $\varepsilon_{\mathcal{F},m}^m$-neighborhood (or tubular neighborhood) of $\mathcal{F}_m$,

$$ \mathcal{D}_{\mathcal{F}_m} (\varepsilon_{\mathcal{F},m}^m) = \{ M \in \mathbb{R}^2, d(M, \mathcal{F}_m) \le \varepsilon_{\mathcal{F},m}^m \}, \quad (\mathbb{R} 98) $$

of tubular volume $V_{m,\mathcal{F}_m} (\varepsilon_{\mathcal{F},m}^m)$, we define, much as in [LRŽ17b], the lower r-dimensional Minkowski content (resp., the upper r-dimensional Minkowski content) of the IFD as

$$ M_{\star}^r (\mathcal{F}^I) = \liminf_{m \to \infty} \frac{V_{m,\mathcal{F}_m} (\varepsilon_{\mathcal{F},m}^m)}{(\varepsilon_{\mathcal{F},m}^m)^{2-r}} \quad (\text{resp., } M^{\star,r} (\mathcal{F}^I) = \limsup_{m \to \infty} \frac{V_{m,\mathcal{F}_m} (\varepsilon_{\mathcal{F},m}^m)}{(\varepsilon_{\mathcal{F},m}^m)^{2-r}}) \cdot \quad (\mathbb{R} 99) $$

Recall that $\lim_{m \to \infty} \varepsilon_{\mathcal{F},m}^m = 0$; see Definition 3.2, on page 38, along with Definition 3.1, on page 37, for the special case of the Weierstrass IFD, for which we also have (in the present notation),

$$ V_{m,\mathcal{F}_m} (\varepsilon_{\mathcal{F},m}^m) = \tilde{V}_{m,\mathcal{F}_m} (\varepsilon_{\mathcal{F},m}^m), $$

for all $m \in \mathbb{N}$.
Note that, by definition, we have that

$$0 \le \mathcal{M}_{\star}^{r}(\mathcal{F}^{\mathcal{I}}) \le \mathcal{M}^{*,r}(\mathcal{F}^{\mathcal{I}}) \le \infty. \qquad (\mathcal{R} 100)$$

We then define the lower Minkowski dimension (resp., the upper Minkowski dimension) of the IFD by

$$\underline{D}(\mathcal{F}^{\mathcal{I}}) = \inf \{r \ge 0, \mathcal{M}_{\star}^{r}(\mathcal{F}^{\mathcal{I}}) < \infty\} \qquad (\mathcal{R} 101)$$

$$(\text{resp., } \overline{D}(\mathcal{F}^{\mathcal{I}}) = \inf \{r \ge 0, \mathcal{M}^{*,r}(\mathcal{F}^{\mathcal{I}}) < \infty\}). \qquad (\mathcal{R} 102)$$

As usual, by definition, the Minkowski dimension $D_{\mathcal{F}^{\mathcal{I}}} = D(\mathcal{F}^{\mathcal{I}})$ of the IFD exists if

$$\underline{D}(\mathcal{F}^{\mathcal{I}}) = \overline{D}(\mathcal{F}^{\mathcal{I}}), \qquad (\mathcal{R} 103)$$

in which case, of course, we have that

$$D_{\mathcal{F}^{\mathcal{I}}} = D(\mathcal{F}^{\mathcal{I}}) = \underline{D}(\mathcal{F}^{\mathcal{I}}) = \overline{D}(\mathcal{F}^{\mathcal{I}}). \qquad (\mathcal{R} 104)$$

**Definition 4.6 (Minkowski Nondegeneracy and Minkowski Measurability of an IFD).**

Let $\mathcal{F}^{\mathcal{I}}$ be an arbitrary IFD. Assume that its Minkowski dimension $D_{\mathcal{F}^{\mathcal{I}}}$ exists, in the sense of Definition 4.5, on page 88 just above.

Then, with the same notation as in Definition 4.5, the IFD $\mathcal{F}^{\mathcal{I}}$ is said to be *Minkowski nondegenerate* if the lower and upper Minkowski contents,

$$\mathcal{M}_{\star}^{D_{\mathcal{F}^{\mathcal{I}}}}(\mathcal{F}^{\mathcal{I}}) = \liminf_{m \to \infty} \frac{\mathcal{V}_{m, \mathcal{F}_m}(\varepsilon_{\mathcal{F},m}^m)}{(\varepsilon_{\mathcal{F},m}^m)^{2-D_{\mathcal{F}^{\mathcal{I}}}}} \quad \text{and} \quad \mathcal{M}^{*,D_{\mathcal{F}^{\mathcal{I}}}}(\mathcal{F}^{\mathcal{I}}) = \limsup_{m \to \infty} \frac{\mathcal{V}_{m, \mathcal{F}_m}(\varepsilon_{\mathcal{F},m}^m)}{(\varepsilon_{\mathcal{F},m}^m)^{2-D_{\mathcal{F}^{\mathcal{I}}}}}$$

are respectively positive and finite. Recall that the inequalities in (R100) always hold.

Finally, the IFD $\mathcal{F}^{\mathcal{I}}$ is said to be *Minkowski measurable* if it is Minkowski nondegenerate and

$$\mathcal{M}_{\star}^{D_{\mathcal{F}^{\mathcal{I}}}}(\mathcal{F}^{\mathcal{I}}) = \mathcal{M}^{*,D_{\mathcal{F}^{\mathcal{I}}}}(\mathcal{F}^{\mathcal{I}}); \qquad (\mathcal{R} 105)$$

i.e., if the following limit exists in $]0, +\infty[$ (and necessarily equals this common value, denoted by $\mathcal{M}^{D_{\mathcal{F}^{\mathcal{I}}}}(\mathcal{F}^{\mathcal{I}})$):

$$\mathcal{M}^{D_{\mathcal{F}^{\mathcal{I}}}}(\mathcal{F}^{\mathcal{I}}) = \lim_{m \to \infty} \frac{\mathcal{V}_{m, \mathcal{F}_m}(\varepsilon_{\mathcal{F},m}^m)}{(\varepsilon_{\mathcal{F},m}^m)^{2-D_{\mathcal{F}^{\mathcal{I}}}}}. \qquad (\mathcal{R} 106)$$

Then, $\mathcal{M}^{D_{\mathcal{F}^{\mathcal{I}}}}(\mathcal{F}^{\mathcal{I}})$ is called the *Minkowski content* of the IFD.

*Remark 4.10.* As was mentioned in Definition 4.6, on page 89 above, the IFD is said to be *Minkowski nondegenerate* if

$$0 < \mathcal{M}_{\star}^{D_{\mathcal{F}^{\mathcal{I}}}}(\mathcal{F}^{\mathcal{I}}) < \mathcal{M}^{*,D_{\mathcal{F}^{\mathcal{I}}}}(\mathcal{F}^{\mathcal{I}}) < \infty. \qquad (\mathcal{R} 107)$$
Equivalently, the IFD is Minkowski nondegenerate if there exists $d \ge 0$ such that,

$$0 < \mathcal{M}_{\star}^{d}(\mathcal{F}^{\mathcal{I}}) < \mathcal{M}^{\star d}(\mathcal{F}^{\mathcal{I}}), \qquad (\mathcal{R} 108)$$

which implies that the Minkowski dimension $D_{\mathcal{F}^{\mathcal{I}}}$ of the IFD exists and is equal to $d$.

**Definition 4.7 (Average Lower and Upper Minkowski Contents of an IFD).**

We hereafter use the same notation as in Definition 4.5, on page 88, and in Definition 4.6, on page 89 just above, where $\mathcal{F}^{\mathcal{I}}$ denotes an arbitrary iterated fractal drum of $\mathbb{R}^2$.

Then, by analogy with what can be found in [LRŽ17b], Definition 2.4.1, on page 178, we define, for all $m \in \mathbb{N}$ sufficiently large, the $m^{th}$ effective average lower-dimensional Minkowski content (resp., the $m^{th}$ effective average upper-dimensional Minkowski content) of $\mathcal{F}_m$ as

$$\widetilde{\mathcal{M}}_{\star}^{D_m, e}(\mathcal{F}_m) = \liminf_{r \to +\infty} \frac{1}{\ln r} \int_{\frac{1}{r}}^{\varepsilon_{\mathcal{F},m}^m} t^{D_m-3} \tilde{\mathcal{V}}_{m,\mathcal{F}_m}(t) dt \qquad (\mathcal{R} 109)$$

$$\left( \text{resp., } \widetilde{\mathcal{M}}^{\star, D_m, e}(\mathcal{F}_m) = \limsup_{r \to +\infty} \frac{1}{\ln r} \int_{\frac{1}{r}}^{\varepsilon_{\mathcal{F},m}^m} t^{D_m-3} \tilde{\mathcal{V}}_{m,\mathcal{F}_m}(t) dt \right), \qquad (\mathcal{R} 110)$$

where $\tilde{\mathcal{V}}_{m,\mathcal{F}_m}$ is the natural volume extension of $\mathcal{F}^{\mathcal{I}}$ (or $m^{th}$ effective tubular volume of $\mathcal{F}_m$; see Notation 11, on page 64, along with Definition 4.3, on page 62), and where $D_m$ denotes the abscissa of convergence of the $m^{th}$ local effective tube zeta function $\tilde{\zeta}_{m,\mathcal{F}_m}^e$.

In the case when both of these values coincide, their common value, denoted by $\widetilde{\mathcal{M}}^{D_m, e}(\mathcal{F}_m)$, is called the $m^{th}$ local effective average Minkowski content of $\mathcal{F}_m$, which is then said to exist. Accordingly,

$$\widetilde{\mathcal{M}}^{D_m, e}(\mathcal{F}_m) = \lim_{r \to +\infty} \frac{1}{\ln r} \int_{\frac{1}{r}}^{\varepsilon_{\mathcal{F},m}^m} t^{D_m-3} \tilde{\mathcal{V}}_{m,\mathcal{F}_m}(t)(t) dt \cdot \qquad (\mathcal{R} 111)$$

We can now state several new geometric consequences of our above results, especially, Theorem 4.7, on page 72 and Theorem 4.11, on page 83.

**Theorem 4.12 (Lower, Upper and Average $D_W$-Dimensional Minkowski Contents of the Weierstrass IFD).**

For any $m \in \mathbb{N}$, let us denote by $D_m$ the abscissa of convergence of the $m^{th}$ local effective tube zeta function $\tilde{\zeta}_{m,W}^e$. Then, the Minkowski dimension of the Weierstrass IFD $\Gamma_W^{\mathcal{I}}$ exists and equals $D_m = D_W$, for any sufficiently large $m \in \mathbb{N}^*$, where $D_W = 2 - \ln_{N_b} \frac{1}{\lambda} \in ]1, 2[$ is the Minkowski dimension of the Weierstrass Curve; see Theorem 4.8, on page 76 above. Moreover, the lower and upper $D_W$-dimensional Minkowski contents of the Weierstrass IFD $\Gamma_W^{\mathcal{I}}$, respectively

$$\mathcal{M}_{\star}^{D_m}(\Gamma_W^{\mathcal{I}}) = \mathcal{M}_{\star}^{D_W}(\Gamma_W^{\mathcal{I}}) \text{ and } \mathcal{M}^{\star, D_m}(\Gamma_W^{\mathcal{I}}) = \mathcal{M}^{\star, D_W}(\Gamma_W^{\mathcal{I}}),$$

take strictly positive and finite values; more specifically, they are such that

$$0 < \frac{C_{\text{Rectangles}}}{N_b} < \mathcal{M}_{\star}^{D_m}(\Gamma_W^{\mathcal{I}}) < \mathcal{M}^{\star, D_m}(\Gamma_W^{\mathcal{I}}) \le C_{\text{Rectangles}} < \infty, \qquad (\mathcal{R} 112)$$
where $C_{\text{Rectangles}}$ denotes the strictly positive and finite constant introduced in Property 4.3, on page 65.

Recall that $C_{\text{Rectangles}}$ may depend on $m \in \mathbb{N}^*$, but is uniformly bounded away from 0 and infinity (with bounds independent of $m \in \mathbb{N}^*$ large enough). Hence, the same is true of

$$ \mathcal{M}_{\star}^{D_m}(\Gamma_W^I) = \mathcal{M}_{\star}^{D_W}(\Gamma_W^I) \quad \text{and} \quad \mathcal{M}^{\star, D_m}(\Gamma_W^I) = \mathcal{M}^{\star, D_W}(\Gamma_W^I), $$

where $D_m = D_W$, for all sufficiently large $m \in \mathbb{N}^*$.

In addition, the values of $\mathcal{M}_{\star}^{D_W}(\Gamma_W^I)$ and $\mathcal{M}^{\star, D_W}(\Gamma_W^I)$ are respectively equal to the minimum and maximum value of the one-periodic function $G_{D_W} = G_{0,D_W}$ introduced in Theorem 4.11, on page 83, associated to $D_m$ in the expression of the fractal tube formula given in the same theorem (recall that the periodicity is with respect to the variable $\ln_{N_b}(\varepsilon_m^m)^{-1}$, see Property 4.1, on page 61).

Finally, for all sufficiently large $m \in \mathbb{N}^*$, the $m^{\text{th}}$ local effective average Minkowski content exists and is given by the mean value of the one-periodic function $G_{D_m} = G_{D_W}$, as well as by the residues of $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ at $s = D_m = D_W$:

$$ \tilde{\mathcal{M}}^{D_m, e}(\Gamma_{W_m}) = \int_0^1 G_{D_W}(x) dx = \operatorname{res}(\tilde{\zeta}_{m, \Gamma_{W_m}}^e, D_m) = \frac{\operatorname{res}(\zeta_{m, \Gamma_{W_m}}^e, D_m)}{2 - D_m}. \quad (\mathbb{R} 113) $$

Hence, $\tilde{\mathcal{M}}^{D_m, e}(\Gamma_{W_m})$ is nontrivial; in fact,

$$ 0 < \mathcal{M}_{\star}^{D_m}(\Gamma_W^I) < \tilde{\mathcal{M}}^{D_m, e}(\Gamma_{W_m}) < \mathcal{M}^{\star, D_m}(\Gamma_W^I) < \infty. $$

More specifically, still for all $m$ large enough and thus, with $D_m = D_W$, the $m^{\text{th}}$ local effective average Minkowski content $\tilde{\mathcal{M}}^{D_m, e}(\Gamma_{W_m})$ may depend on $m \in \mathbb{N}^*$, but is uniformly bounded away from 0 and $\infty$ (with bounds independent of $m \in \mathbb{N}^*$ large enough).

**Proof.** In light of Theorem 4.7, on page 72 (and of Definition 4.5, on page 88), one has

$$ \begin{align*} \mathcal{M}^{\star, D_m}(\Gamma_W^I) &= \limsup_{m \to \infty} \left\{ \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}} f_{k, \ell, \text{Rectangles}} (\varepsilon_m^m)^{k(2-D_W)-i\ell \mathbf{p}} \right. \\ &\quad + (\varepsilon_m^m)^{D_W} \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}} \left\{ f_{k, \ell, \text{wedges}, 1} (\varepsilon_m^m)^{1-i\ell \mathbf{p}} + f_{k, \ell, \text{wedges}, 2} (\varepsilon_m^m)^{-1+2k-i\ell \mathbf{p}} + f_{k, \ell, \text{wedges}, 3} (\varepsilon_m^m)^{3+2k-i\ell \mathbf{p}} \right\} \\ &\quad \left. + (\varepsilon_m^m)^{D_W} \sum_{\ell \in \mathbb{Z}, k \in \mathbb{N}} f_{k, \ell, \text{triangles, parallelograms}} (\varepsilon_m^m)^{-i\ell \mathbf{p}} + (\varepsilon_m^m)^{D_W} \pi - (\varepsilon_m^m)^{D_W} \frac{\pi (\varepsilon_m^m)^2}{2} \right\} \\ &= \limsup_{m \to \infty} \sum_{\ell \in \mathbb{Z}} f_{m, 0, \text{Rectangles}} (\varepsilon_m^m)^{-i\ell \mathbf{p}} \\ &= \limsup_{m \to \infty} C_{\text{Rectangles}} \frac{N_b - 1}{N_b} \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i\ell \mathbf{p}}}{\ln N_b + 2i\ell\pi} (\varepsilon_m^m)^{-i\ell \mathbf{p}} = \limsup_{x \to +\infty} C_{\text{Rectangles}} N_b^{-\{x\}}. \tag{R 114} \end{align*} $$

In the same way,

$$ \mathcal{M}_{\star}^{D_W}(\Gamma_W^I) = \liminf_{x \to +\infty} C_{\text{Rectangles}} N_b^{-\{x\}}. \quad (\mathbb{R} 115) $$