The topological property of the irregular sets on the lengths of basic intervals in beta-expansions

Let $\beta >1$ be a real number and $(\epsilon_1(x, \beta), \epsilon_2(x, \beta), \ldots)$bethe$\beta$-expansionofapoint$x \in (0, 1]$\mathrm{.}Forall$x \in (0,1]$,let$A(D(x))$bethesetofaccumulationpointsof$\frac{-\log_\beta |I_n(x)|}{n}$as$n \rightarrow \infty$,where$|I_n(x)|$ is the length of the basic interval of order $n$ containing $x\in (0,1]$. In this paper, we prove that $A(D(x))$ is always a closed interval for any $x \in (0,1]$\mathrm{.}Furthermore,if$\lambda(\beta)>0$, the extremely irregular set containing points $x\in [0,1]$ whose upper limit of $\frac{-\log_\beta |I_n(x)|}{n}$equalsto$1+\l(\beta)$isresidual,where$1+\l(\beta)$ is a constant depending on $\beta$. As a consequence, the irregular set with $x\in [0, 1]$whoselimitof$\frac{-\log_\beta |I_n(x)|}{n}$ does not exist is residual for every $\lambda (\beta )>0$.