Thanks to Property 4.2, on page 62, and since $0 \le \{x\} < 1$, where $\{x\}$ denotes the fractional part of $x \in \mathbb{R}$, we have that

$$N_b^{-\{x\}} = \frac{N_b - 1}{N_b} \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i \ell \mathbf{P}} (\varepsilon_m^m)^{-i \ell \mathbf{P}}}{\ln N_b + 2i \ell \pi}, \quad \text{with } x = -\ln_{N_b}((N_b - 1)\varepsilon_m^m), \quad (\mathbb{R} \ 116)$$

This yields $\frac{1}{N_b} < N_b^{-\{x\}} \le 1$, and thus, in light of Theorem 4.8, on page 76, and with $D_m = D_W$ given as in the theorem, we have that, for all $m \in \mathbb{N}^*$ large enough,

$$\frac{C_{\text{Rectangles}}}{N_b} < \mathcal{M}_{\star}^{D_m} (\Gamma_W^{\mathcal{I}}) < \mathcal{M}^{\star, D_m} (\Gamma_W^{\mathcal{I}}) \le C_{\text{Rectangles}}. \quad (\mathbb{R} \ 117)$$

The constant $C_{\text{Rectangles}}$ being strictly positive and finite (see Property 4.3, on page 65), this accounts for a strictly positive (resp., finite) value of the lower (resp., upper) Minkowski content $\mathcal{M}_{\star}^{D_m} (\Gamma_W^{\mathcal{I}})$ (resp., $\mathcal{M}^{\star, D_m} (\Gamma_W^{\mathcal{I}})$).

Also, still for all $m \in \mathbb{N}$ sufficiently large, the one-periodic function (with respect to the variable $\ln_{N_b}(\varepsilon_m^m)^{-1}$, see Property 4.1, on page 61),

$$G_{D_W} = G_{0,D_W} : \quad x \mapsto \frac{N_b - 1}{N_b} C_{\text{Rectangles}} \sum_{\ell \in \mathbb{Z}} \frac{(N_b - 1)^{-i \ell \mathbf{P}} (\varepsilon_m^m)^{-i \ell \mathbf{P}}}{\ln N_b + 2i \ell \pi} = N_b^{-\{x\}}, \quad (\mathbb{R} \ 118)$$

associated to the value $D_W = D_m$ is nonconstant, because it has nonzero $m^{th}$ Fourier coefficients, with $m \ne 0$, as can be seen from the fractal tube formula, and as stated in Theorem 4.11, on page 83. (Note that the function $G_{D_W} = G_{D_m}$ may depend on $m$ sufficiently large.)

The last part of the theorem, regarding the $m^{th}$ local effective average Minkowski content $\tilde{M}^{D_{m,e}}(\Gamma_{W_m})$ of the Weierstrass IFD (as introduced in Definition 4.7, on page 90), follows at once from the method of proof of [LRŽ17b], Theorem 2.3.25, on page 157. Note that the fact that $\tilde{M}^{D_{m,e}}(\Gamma_{W_m})$ is uniformly bounded away from 0 and infinity (in $m \in \mathbb{N}^*$ large enough) follows from relation (R88) on page 82. Indeed, recall from Property 4.3 on page 65 that the coefficients $f_{k,\ell,\text{Rectangles}}$ are uniformly bounded away from 0 and infinity (with bounds independent of $m \in \mathbb{N}^*$ large enough).

□

**Corollary 4.13 (of Theorem 4.12) Minkowski Dimension – Minkowski Nondegeneracy).**

The Weierstrass IFD is Minkowski nondegenerate. Furthermore, the number $D_W = 2 - \ln_{N_b} \frac{1}{\lambda}$ is a simple Complex Dimension of the IFD, and it coincides with the Minkowski Dimension of $\Gamma_W$, which must also exist. Moreover, the Weierstrass IFD is not Minkowski measurable.

*Proof.* In light of Theorem 4.12, on page 90, the nondegeneracy directly follows from the definition. The statement concerning $D_m = D_W$ (for all $m \in \mathbb{N}$ sufficiently large) then follows from Definition 4.6, on page 89, in particular.
Furthermore, the Weierstrass IFD is not Minkowski measurable; i.e., here,

$$ \mathcal{M}_{\star}^{D_m}(\Gamma_W^I) < \mathcal{M}^{\star, D_m}(\Gamma_W^I). $$

This last statement also follows from Theorem 4.12, on page 90, because the one-periodic function $G_{D_W} = G_{D_m}$ is nonconstant, and so, by the method of proof of the results in [LRŽ17b], Theorem 2.3.25, on page 157,

$$ \tilde{\mathcal{M}}_{\star}^{D_m, e}(\Gamma_{W_m}) = \min_{[0,1]} G_{D_W} < \max_{[0,1]} G_{D_W} = \tilde{\mathcal{M}}^{\star, D_m, e}(\Gamma_{W_m}). \quad (\mathbb{R} \ 119) $$

Moreover, since, for all $m \in \mathbb{N}$ sufficiently large, the $m^{th}$ local effective distance zeta function $\zeta_{m, \Gamma_{W_m}}^e$ associated to the Weierstrass IFD can clearly be meromorphically extended to a connected neighborhood of $s = D_W$ in the Complex Plane, $D_W$ is a simple pole of $\zeta_{m, \Gamma_{W_m}}^e$. As was pointed out at the end of Theorem 4.12, given on page 90, in agreement with the general theory in [LRŽ17b] (see Theorem 2.3.25, page 157). $\square$

**Remark 4.11.** Let us call the global lower (resp., upper) effective average Minkowski content of the Weierstrass IFD $\Gamma_W^I$, and denote by $\tilde{\mathcal{M}}_{\star}^{D_W, e}(\Gamma_W^I)$ (resp., $\tilde{\mathcal{M}}_{\star}^{D_W, e}(\Gamma_W^I)$) the following lower (resp., upper) limit of the corresponding $m^{th}$ local effective average Minkowski contents, with $D_m = D_W$, for all $m \in \mathbb{N}^*$ sufficiently large:

$$ \tilde{\mathcal{M}}_{\star}^{D_W, e}(\Gamma_W^I) = \liminf_{m \to \infty} \tilde{\mathcal{M}}_{\star}^{D_m, e}(\Gamma_{W_m}) \quad (\mathbb{R} \ 120) $$

$$ \left( \text{resp., } \tilde{\mathcal{M}}_{\star}^{D_W, e}(\Gamma_W^I) = \limsup_{m \to \infty} \tilde{\mathcal{M}}_{\star}^{D_m, e}(\Gamma_{W_m}) \right) $$

Then, it follows from Theorem 4.12, on page 90, that the above quantities are well defined and bounded away from 0 and $\infty$. Furthermore, they coincide; so that the global effective average Minkowski content of the Weierstrass IFD $\Gamma_W^I$, denoted by $\tilde{\mathcal{M}}_{\star}^{D_W, e}(\Gamma_W^I)$, exists.

In light of relation (R112), and since $D_m = D_W$, for all $m \in \mathbb{N}^*$ sufficiently large, we obtain that

$$ 0 < \frac{C_{\text{Rectangles}}}{N_b} < \mathcal{M}_{\star}^{D_W}(\Gamma_W^I) \le \tilde{\mathcal{M}}_{\star, D_W, e}(\Gamma_{W_m}) \le \mathcal{M}_{\star}^{D_W}(\Gamma_W^I) \le C_{\text{Rectangles}} < \infty. \quad (\mathbb{R} \ 121) $$

In addition, since $D_m = D_W$, and, by relation (R113),

$$ \tilde{\mathcal{M}}_{\star}^{D_W, e}(\Gamma_{W_m}) = \text{res} \left( \tilde{\zeta}_{m, \Gamma_{W_m}}^e, D_W \right) = \frac{\text{res} \left( \zeta_{m, \Gamma_{W_m}}^e, D_W \right)}{2 - D_W}, \quad (\mathbb{R} \ 122) $$

for all $m \in \mathbb{N}^*$ sufficiently large, as well as (see Theorem 4.8, on page 76, and its proof),

$$ \text{res} \left( \tilde{\zeta}_{\Gamma_W}^e, D_W \right) = \lim_{m \to \infty} \text{res} \left( \tilde{\zeta}_{m, \Gamma_{W_m}}^e, D_W \right), \quad (\mathbb{R} \ 123) $$

which follows from the local uniform convergence (as $m \to \infty$) on $\mathbb{C}$ of $\tilde{\zeta}_{m, \Gamma_{W_m}}^e$ (resp., $\zeta_{m, \Gamma_{W_m}}^e$) to $\tilde{\zeta}_{\Gamma_W}^e$ (resp., to $\zeta_{\Gamma_W}^e$).

By combining relation (R122) and relation (R123), we see that $\tilde{\mathcal{M}}_{\star}^{D_W, e}(\Gamma_W^I)$ exists and satisfies

$$ \tilde{\mathcal{M}}_{\star}^{D_W, e}(\Gamma_W^I) = \lim_{m \to \infty} \mathcal{M}_{\star, D_W, e}(\Gamma_{W_m}) = \text{res} \left( \tilde{\zeta}_{\Gamma_W}^e, D_W \right) = \frac{\text{res} \left( \zeta_{\Gamma_W}^e, D_W \right)}{2 - D_W}. \quad (\mathbb{R} \ 124) $$
Finally, in light of relation (R121), we deduce from relation (R124) that

$$0 < \frac{1}{N_b} \liminf_{m \to \infty} C_{\text{Rectangles}} \le \mathcal{M}^*_{\star} D_W (\Gamma_W^I) \le \tilde{\mathcal{M}}^{D_{W,e}} (\Gamma_W^I) \le \mathcal{M}^{*D_W} (\Gamma_W^I) \le \limsup_{m \to \infty} C_{\text{Rectangles}} < \infty. \quad (\mathcal{R} 125)$$

In conclusion, the global effective average Minkowski content $\tilde{\mathcal{M}}^{D_{W,e}}(\Gamma_W^I)$ of the Weierstrass IFD $\Gamma_W^I$, exists, is positive and finite, satisfies the estimates in relation (R125), and is expressed via relation (R124) in terms of the residues at $s = D_W$ of the global effective tube and distance zeta functions of $\Gamma_W^I$.

Accordingly, in particular, the relation between the $m^{th}$ local effective average Minkowski content and the residues at $s = D_W$ of the $m^{th}$ local effective tube and distance zeta functions, for all $m \in \mathbb{N}^*$ sufficiently large (see relation (R113), on page 91) remains precisely the same between their global counterparts.

## 4.5 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

$$W(x) = \sum_{n=0}^{\infty} \lambda^n \cos(2\pi b^n x), \quad (\mathcal{R} 126)$$

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, $T_W = \{T_i\}_{i \in \mathbb{Z}}$, such that, for any integer $i$ and $(x, y)$ in $\mathbb{R}^2$,

$$T_i(x, y) = \left( \frac{x + i}{b}, \lambda y + \cos \left( 2\pi \left( \frac{x + i}{b} \right) \right) \right), \quad (\mathcal{R} 127)$$

it is natural to consider the associated *infinite IFS (IIFS)*, $T_W$. As a consequence, the resulting prefractal graphs are infinite ones.

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 2.11, on page 24, and Corollary 2.12, on page 24, and, consequently, to the resulting ones about the elementary heights obtained in Corollary 2.15, on page 27.

As for the effective tubular neighborhood, due to the polygonal approximations induced 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 Proposition 3.4, on page 50. 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 4.7, on page 72, where the powers of the small parameter $\varepsilon_m^m$ would be, respectively, and as previously,

$$ (\varepsilon_m^m)^{2-D_W+k(2-D_W)-i\ell\mathbf{p}}, \quad (\varepsilon_m^m)^{3-i\ell\mathbf{p}}, \quad (\varepsilon_m^m)^{1+2k-i\ell\mathbf{p}}, \quad (\varepsilon_m^m)^{5+2k-i\ell\mathbf{p}}, \quad (\varepsilon_m^m)^{2-i\ell\mathbf{p}}, \quad (\varepsilon_m^m)^2, \quad (\mathbb{R}^{128}) $$

which would yield the same results concerning the possible Complex Dimensions, along with the upper and lower, as well as the average, Minkowski contents of the Weierstrass Curve.

As in the integer case, the terms involving $(\varepsilon_m^m)^{2-D_W+k(2-D_W)-i\ell\mathbf{p}}$ come from the contribution of the rectangles. The one-periodic functions (with respect to the variable $\ln_b(\varepsilon_m^m)^{-1}$ this time), respectively associated to the values $D_W - k(2 - D_W)$, $k \in \mathbb{N}$, are thus nonconstant, with all of their Fourier coefficients being nonzero. Hence, as in Theorem 4.10, on page 81, for each $k \in \mathbb{N}$ and $\ell \in \mathbb{Z}$, $D_W - k(2 - D_W) + i\ell\mathbf{p}$, are all simple Complex Dimensions of the Weierstrass Curve; i.e., they are simple poles of the tube (or, equivalently, of the distance) zeta function.

We also mention that we could deal with the case $\lambda b < 1$, exactly in the same manner, and with the same conclusions. Actually, it is noteworthy that, in the present paper, all of our results remain valid 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 $C^1$, 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 tube and distance 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 $D_W$. According to [LRŽ17b], part c. of Theorem 2.1.11, page 57, and the last statement of Theorem 2.2.11, page 121, this additional result follows from the fact that, for all $m \in \mathbb{N}$ sufficiently large, $D_m = D_W$ exists, $M_{\star}^{D_W}(\Gamma_W^T) > 0$ and $D_W < 2$. It can also be deduced from Theorem 4.7, on page 72, or else from Theorem 4.10, on page 81.

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 prefractal approximations. In [DL23b], by means of the exact sequence of the local effective 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 – the one associated with the limit fractal object – has the same poles as the fractal zeta function at each step of the prefractal approximation, and, hence, that the Complex Dimensions of the fractal are the same as the Complex Dimensions of each prefractal approximation. As is shown in [DL23b], the determination of the explicit Complex Dimensions of the IFD is a compulsory step in order to obtain the Complex Dimensions of the limit fractal Curve.

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, [DL22c], where we determine the fractal cohomology of the Weierstrass Curve.

Behind the fractal series expansion of the Weierstrass function, another expansion, indexed by the Complex Dimensions obtained in our fractal tube formulas (see Theorem 4.7, on page 72 and
Theprem 4.11, on page 83 above), naturally arises. 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 differentiation could be achieved on this, however, everywhere singular object; see [DL22a] and [DL22c].

As was mentioned in Subsection 4.5, on page 94, we also intend, in a future work, to extend our results to the general case, i.e., when the Weierstrass function $\mathcal{W}$ is defined, for any real number $x$, by

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

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 [Dav19]), including the Takagi function, the Knopp functions and the Koch parametrized Curve; see [DL23a].

The reader may wonder where there is an intrinsic way of obtaining the global fractal zeta functions introduced and studied in Theorem 4.8 and Corollary 4.9 (on pages 76 and 80, respectively), that would be more in keeping with the general theory of Complex Dimensions (as developed in [LRŽ17a]–[LRŽ17c] and [LRŽ18]) and its natural extensions (e.g., in [LW23]). This question is addressed by the authors in [DL23b], by using the polyhedral measure introduced in [DL22b].

## Acknowledgements

The authors would like to thank Goran Radunović and Steffen Winter for their helpful questions about the first version of this paper.

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