Level sets of the run-length function of beta-expansions

Lixuan Zheng

Published 2018-05-12Version 1

For any $\beta > 1$, denoted by $r_n(x,\beta)$ the maximal length of consecutive zeros amongst the first $n$ digits of the $\beta$-expansion of $x\in[0,1)$. Let $0\leq a\leq b\leq 1$. The Hausdorff dimension of the level set $$E_{a,b}=\left\{x \in [0,1): \liminf_{n\rightarrow \infty}\frac{r_n(x,\beta)}{n}=a,\ \limsup_{n\rightarrow \infty}\frac{r_n(x,\beta)}{n}=b\right\}$$ is obtained. As a consequence, the set $$E_a=\left\{x \in [0,1):\liminf_{n\rightarrow \infty}\frac{r_n(x,\beta)}{n}=a\right\}$$ has Hausdorff dimension ${(1-2a)}^2$ when $0\leq a\leq\frac{1}{2}$. We show that the extremely divergent set $E_{0,1}$ which is of zero Hausdorff dimension is, however, residual which means that it is large from the topological viewpoint. The same problems in the parameter space are also examined.