Better estimates of H\"older thickness of fractals

Abstract

Dimensions of level sets of generic continuous functions and generic H\"older functions defined on a fractal $F$ encode information about the geometry, ``the thickness" of $F$. While in the continuous case this quantity is related to a reasonably tame dimension notion which is called the topological Hausdorff dimension of $F$, the H\"older case seems to be highly nontrivial. A number of earlier papers attempted to deal with this problem, carrying out investigation in the case of Hausdorff dimension and box dimension. In this paper we continue our study of the Hausdorff dimension of almost every level set of generic $1$-H\"older-$ {\alpha}$ functions, denoted by $D_{*}( {\alpha}, F)$. We substantially improve previous lower and upper bounds on $D_{*}( {\alpha}, \Delta)$, where $\Delta$ is the Sierpi\'nski triangle, achieving asymptotically equal bounds as $\alpha\to 0+$. Using a similar argument, we also give an even stronger lower bound on the generic lower box dimension of level sets. Finally, we construct a connected fractal $F$ on which there is a phase transition of $D_{*}( {\alpha}, F)$, thus providing the first example exhibiting this behavior.