Hausdorff dimensions of sets related to Erd\"{o}s-R\'{e}nyi averages in beta expansions

Abstract

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