Hausdorff dimensions of sets related to Erdös-Rényi average in beta expansions

Haibo Chen

Published 2018-04-18Version 1

Let $\beta>1$ and $\phi$ be an integer function defined on $\mathbb{N}\setminus\{0\}$ satisfying $1\leq\phi(n)\leq n$. Define the level set \begin{align*} ER_\phi^\beta(\alpha)=\left\{x\in[0,1]\colon A_\phi(x,\beta)=\alpha\right\},\quad \alpha\in I_\beta, \end{align*} where $A_\phi(x,\beta)$ is the Erd\"{o}s-R\'{e}nyi digit average of $x\in[0,1]$ associated with the function $\phi$ in $\beta$-expansion and $I_\beta$ is the range of $\alpha$. In this paper, we will determine the Hausdorff dimension of $ER_\phi^\beta(\alpha)$ under the assumption $\phi(n)\to\infty$ as $n\to\infty$ and $\phi$ is the integer part of a slowly varying sequence. Besides, some generalizations in $\beta$-expansion to the classic work \cite{Be} of Besicovitch are also given.