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

TL;DR

This paper derives an explicit formula for the upper box dimension of generic H"older level sets on a broad class of self-similar fractals, advancing understanding of their geometric properties.

Contribution

It provides the first explicit formula for the upper box dimension of generic H"older level sets on self-similar fractals, extending previous bounds.

Findings

Almost all level sets have upper box dimension equal to im_H F - lpha

The result applies to a large class of self-similar sets

The formula is valid for generic 1-Hf6lder-b1b7 functions

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 $\dim_H F - \alpha$.