Diophantine approximation and run-length function on β-expansions

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 $x\in[0,1]$. The limit superior (respectively limit inferior) of $\frac{r_n(x,\beta)}{n}$ is linked to the classical Diophantine approximation (respectively uniform Diophantine approximation). We obtain the Hausdorff dimension of the level set $$E_{a,b}=\left\{x \in [0,1]: \liminf_{n\rightarrow \infty}\frac{r_n(x,\beta)}{n}=a,\ \limsup_{n\rightarrow \infty}\frac{r_n(x,\beta)}{n}=b\right\}\ (0\leq a\leq b\leq1).$$ Furthermore, we show that the extremely divergent set $E_{0,1}$ which is of zero Hausdorff dimension is, however, residual. The same problems in the parameter space are also examined.