Level sets of prevalent Weierstrass functions

Abstract

The ‐Weierstrass function is defined as , where is a Lipschitz function on the unit circle. For a prevalent ‐Weierstrass function, we prove that the upper Minkowski dimension of every level set is at most , and the Hausdorff dimension of almost every level set equals with respect to its occupation measure. We further demonstrate that the occupation measure of a prevalent ‐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 , we construct Lipschitz functions such that the mapping is ‐bi‐Hölder. We also prove that such an embedding requires at least coordinate functions.