Exact formula on upper box dimension of generic Hölder level sets
Zoltán Buczolich, Balázs Maga
TL;DR
The article resolves the upper box-dimension of generic level sets for a broad class of Hölder-regular functions on connected self-similar sets with finitely many directions, proving that if $\dim_H F = s>1$ then for generic $f\in C_1^{\alpha}(F)$ and Lebesgue almost every $r\in f(F)$, $\overline{\dim}_B f^{-1}(r) = s-\alpha$. The approach blends existing dimension bounds with a new two-parameter family of auxiliary functions $\phi_{n,m}$ to control level-set dimensions, plus a dense $G_\delta$ argument adapted to locally connected fractals. A key contribution is the construction of dense piecewise affine approximations that preserve Hölder regularity while ensuring level sets realize the target dimension, enabling a sharp generic formula. The results illuminate the geometry of level sets under Hölder regularity on self-similar sets and open avenues for extensions to smaller dimensions and to broader connectivity regimes.
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 Hölder 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-Hölder-$α$ function defined on $F\subseteq \mathbb{R}^p$ has upper box dimension $\dim_H F - α$.