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