Exact formula on upper box dimension of generic H\"older level sets

Abstract

In the previous decades, the size of level sets of functions have been extensively studied in various setups involving different regularity properties and size notions. In the case of H\"older functions, the authors have provided various bounds, but to date no explicit formulae have been found for any studied dimension and the results were valid only about very specific fractals. In this paper, for the first time, we have a result valid for a large class of self-similar sets, namely we prove that for these fractals Lebesgue almost every level set of the generic 1-H\"older-$\alpha$ function defined on $F\subseteq \mathbb{R}^p$ has upper box dimension $\dim_H F - \alpha$.