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 $p\geq2$ 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 $0\leq\alpha\leq\beta\leq p-1$.