Hausdorff dimensions of level sets related to moving digit means

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 $p\ge 2$ is an integer. Moreover, the Hausdorff dimension of level set \[B(α,β)=\left\{x\in [0,1]\colon \underline{M}(x)=α,\overline{M}(x)=β\right\}\] is determined for each pair of numbers $\alpha$ and $\beta$ satisfying with $0\le \alpha \le \beta \le p-1$.

15 pages