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

Abstract

Let $\beta >1$, $I$ be the unite interval $[0,1)$ and $\varphi$ be an integer function defined on $\mathbb{N}\setminus \{0\}$ satisfying $1\le \varphi (n)\le n$. Denote by $A_{\varphi }(x,\beta )$ the Erdös-Rényi average of $x\in I$ associated with the function $\varphi$ in $\beta$-expansion and $I_{\beta }$ the range of $A_{\varphi }(x,\beta )$ for $x\in I$. For the level set \begin{align*} ER_ϕ^β(α)=\left\{x\in I\colon A_ϕ(x,β)=α\right\},\quad\text{where}\ α\in I_β, \end{align*} in this paper we will determine its Hausdorff dimension under the assumption $\varphi (n)\to \infty$ as $n\to \infty$ and $\varphi$ 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.

25 pages