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

TL;DR

This paper investigates the Hausdorff dimensions of certain level sets related to Erdős-Rényi averages in beta expansions, extending classical results and providing explicit dimension formulas under specific growth conditions on the averaging function.

Contribution

It determines the Hausdorff dimension of Erdős-Rényi average level sets in beta expansions for functions tending to infinity, generalizing Besicovitch's classical work.

Findings

Explicit Hausdorff dimension formulas derived

Dimension results hold for slowly varying functions

Generalization of Besicovitch's classical work

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.