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

Abstract

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

18 pages