# The exceptional sets on the run-length function of beta-expansions

## TL;DR

This paper studies the size and structure of the set of points in [0,1] whose run-lengths in beta-expansions fluctuate wildly relative to a growing function, revealing conditions under which this set is large or negligible.

## Contribution

It characterizes the Hausdorff dimension and residuality of exceptional sets in beta-expansions based on the growth rate of a function controlling run-lengths.

## Key findings

- The set is either empty or has full Hausdorff dimension.
- The set is residual in [0,1] under certain growth conditions.
- The size of the set depends on the rate at which the function vries.

## Abstract

Let $\beta > 1$ and the run-length function $r_n(x,\beta)$ be the maximal length of consecutive zeros amongst the first n digits in the $\beta$-expansion of $x\in[0,1]$. The exceptional set $$E_{\max}^{\varphi}=\left\{x \in [0,1]:\liminf_{n\rightarrow \infty}\frac{r_n(x,\beta)}{\varphi(n)}=0, \limsup_{n\rightarrow \infty}\frac{r_n(x,\beta)}{\varphi(n)}=+\infty\right\}$$ is investigated, where $\varphi: \mathbb{N} \rightarrow \mathbb{R}^+$ is a monotonically increasing function with $\lim\limits_{n\rightarrow \infty }\varphi(n)=+\infty$. We prove that the set $E_{\max}^{\varphi}$ is either empty or of full Hausdorff dimension and residual in $[0,1]$ according to the increasing rate of $\varphi$ .