  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.
