Hausdorff dimensions of level sets related to moving digit means

TL;DR

This paper investigates the Hausdorff dimensions of level sets defined by lower and upper moving digit means in p-adic expansions, providing a comprehensive dimension formula for these sets based on digit mean bounds.

Contribution

It introduces the concepts of lower and upper moving digit means and determines the Hausdorff dimension of their level sets, extending understanding of digit mean dynamics in p-adic expansions.

Findings

Hausdorff dimension formulas for level sets with specified digit mean bounds

Characterization of the dimension based on the pair (α, β) within [0, p-1]

Extension of digit mean analysis to p-adic expansions

Abstract

In this paper, we will introduce and study the lower moving digit mean $\underline{M}(x)$ and the upper moving digit mean $\overline{M}(x)$ of $x\in[0,1]$ in $p$-adic expansion, where $p\geq2$ is an integer. Moreover, the Hausdorff dimension of level set \[B(\alpha,\beta)=\left\{x\in [0,1]\colon \underline{M}(x)=\alpha,\overline{M}(x)=\beta\right\}\] is determined for each pair of numbers $\alpha$ and $\beta$ satisfying with $0\leq\alpha\leq\beta\leq p-1$.