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Let $\beta > 1$ be a real number and $(\epsilon_1(x, \beta), \epsilon_2(x,\beta), \ldots)$ be the $\beta$-expansion of a point $x \in (0, 1]$. For all $x\in (0,1]$, let $A(D(x))$ be the set of accumulation points of$\frac{-\log_\beta |I_n(x)|}{n}$ as $n \rightarrow \infty$, where $|I_n(x)|$ isthe length of the basic interval of order $n$ containing $x \in (0, 1]$. Inthis paper, we prove that $A(D(x))$ is always a closed interval for any $x \in(0,1]$. Furthermore, if $\lambda(\beta)>0$, the extremely irregular setcontaining points $x \in [0, 1]$ whose upper limit of $\frac{-\log_\beta|I_n(x)|}{n}$ equals to $1+\l(\beta)$ is residual, where $1+\l(\beta)$ is aconstant depending on $\beta$. As a consequence, the irregular set with $x\in[0, 1]$ whose limit of $\frac{-\log_\beta |I_n(x)|}{n}$ does not exist isresidual for every $\lambda(\beta)>0$.

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