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

Abstract

Let $β>1$, $I$ be the unite interval $[0,1)$ and $ϕ$ be an integer function defined on $\mathbb{N}\setminus\{0\}$ satisfying $1\leqϕ(n)\leq n$. Denote by $A_ϕ(x,β)$ the Erdös-Rényi average of $x\in I$ associated with the function $ϕ$ in $β$-expansion and $I_β$ the range of $A_ϕ(x,β)$ 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 $ϕ(n)\to\infty$ as $n\to\infty$ and $ϕ$ is the integer part of some slowly varying sequence. Besides, a generalization to the classic work \cite{Be} of Besicovitch is also given in $β$-expansion.