Level sets of prevalent Weierstrass functions

TL;DR

This paper investigates the fractal dimensions of level sets of prevalent Weierstrass functions, showing that their Hausdorff dimension is almost surely $1-eta$ and that their occupation measures are absolutely continuous.

Contribution

It establishes the Hausdorff dimension of level sets for prevalent Weierstrass functions and introduces a Weierstrass embedding with bi-Hölder properties, requiring at least $1/eta$ functions.

Findings

Hausdorff dimension of level sets is $1-eta$ for prevalent functions

Occupation measure is absolutely continuous with respect to Lebesgue measure

Constructs bi-Hölder embeddings with at least $1/eta$ functions

Abstract

The $\alpha$-Weierstrass function is defined as $W_g^{\alpha,b}(x) = \sum_{k=0}^{\infty} b^{-\alpha k} g(b^k x)$, where $g$ is a Lipschitz function on the unit circle. For a prevalent $\alpha$-Weierstrass function, we prove that the upper Minkowski dimension of every level set is at most $1-\alpha$, and the Hausdorff dimension of almost every level set equals $1-\alpha$ with respect to its occupation measure. We further demonstrate that the occupation measure of a prevalent $\alpha$-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 $x \mapsto \big(W_{g_0}^{\alpha,b}(x), W_{g_1}^{\alpha,b}(x), \ldots, W_{g_{d-1}}^{\alpha,b}(x)\big)$ is $\alpha$-bi-H\"older. We also prove that such an embedding requires at least $1/\alpha$ coordinate functions.