Level sets of prevalent Weierstrass functions

TL;DR

The paper studies level sets of prevalent $α$-Weierstrass functions $W_g^{α,b}(x)=\sum_{k=0}^{∞} b^{-αk} g(b^k x)$ for Lipschitz $g$ on the circle, establishing that every level set has upper Minkowski dimension at most $1-α$, while almost every level set in the sense of the occupation measure has Hausdorff dimension exactly $1-α$; moreover, the occupation measure is absolutely continuous with respect to Lebesgue measure, enabling the transfer of this dimension information to a positive Lebesgue-measure set of level sets. A central tool is the Weierstrass embedding: for sufficiently large dimension $d$, a finite collection of Lipschitz functions $g_0,\,\dots,\,g_{d-1}$ yields an $α$-bi-Hölder embedding $x\mapsto (W_{g_0}^{α,b}(x),\dots,W_{g_{d-1}}^{α,b}(x))$, and almost every perturbation in the corresponding probe space preserves the dimension bound via known prevalence results. The Fourier-analytic analysis of occupation measures $\lambda_t=(W_t)_*\mathcal{L}^1$ (with $W_t=W+\langle t,\Phi\rangle$) shows absolute continuity for almost every perturbation, with $L^2$ densities (and, for $0<α<1/2$, bounded continuous densities), yielding a robust picture of level-set sizes and making the $1-α$ bound sharp in a prevalent sense. The work combines a constructive embedding with occupation-measure and energy methods to advance understanding of the intricate geometry of non-smooth Weierstrass graphs.

Abstract

The $α$-Weierstrass function is defined as $W_g^{α,b}(x) = \sum_{k=0}^{\infty} b^{-αk} g(b^k x)$, where $g$ is a Lipschitz function on the unit circle. For a prevalent $α$-Weierstrass function, we prove that the upper Minkowski dimension of every level set is at most $1-α$, and the Hausdorff dimension of almost every level set equals $1-α$ with respect to its occupation measure. We further demonstrate that the occupation measure of a prevalent $α$-Weierstrass function is absolutely continuous with respect to the Lebesgue measure. Consequently, the result on the Hausdorff dimension of level sets applies to a set of level sets with positive Lebesgue measure. A central tool in our analysis is the Weierstrass embedding. For a sufficiently large dimension $d$, we construct Lipschitz functions $g_0, g_1, \ldots, g_{d-1}$ such that the mapping $x \mapsto \big(W_{g_0}^{α,b}(x), W_{g_1}^{α,b}(x), \ldots, W_{g_{d-1}}^{α,b}(x)\big)$ is $α$-bi-Hölder. We also prove that such an embedding requires at least $1/α$ coordinate functions.

Figures (2)

• Figure 1: The function $h_\ell$ with $\ell=\tfrac{1}{15}$ is depicted on the left and the function $g_i$, where we have chosen $r_i=\tfrac{7}{30}$ and $s_i=\tfrac{9}{30}$ for illustrative purposes, is on the right.
• Figure 2: The blue function in the bottom-left figure represents the Weierstrass function $W_g^{\alpha,b}$, corresponding to the function $g$ shown in the top-left figure. The right-hand figure depicts the zero loci and regions violating \ref{['eq:embedding-goal']} for three functions defined by $(x,y) \mapsto W_{g_i}^{\alpha,b}(y) - W_{g_i}^{\alpha,b}(x)$, where $i \in \{1, 2, 3\}$, and $g_1, g_2, g_3$ are distinct translations of $g$. All three functions are displayed in the bottom-left figure.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 1
• proof
• Theorem 2.1
• proof
• Proposition 1
• proof : Proof of \ref{['thm:main1']}
• Theorem 3.1