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

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 $$E_{\max }^{\phi }=\left\{x\in [0,1]:\underset{n\to \infty }{\mathrm{liminf}}\frac{r_n(x,\beta )}{\phi (n)}=0,\underset{n\to \infty }{\mathrm{limsup}}\frac{r_n(x,\beta )}{\phi (n)}=+\infty \right\}$$ is investigated, where $\phi :\mathbb{N}\to {\mathbb{R}}^{+}$ is a monotonically increasing function with $\underset{n\to \infty }{\lim }\phi (n)=+\infty$. We prove that the set $E_{\max }^{\phi }$ is either empty or of full Hausdorff dimension and residual in $[0,1]$ according to the increasing rate of $\phi$ .

18 pages