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

Abstract

Let $\beta >1$ be a real number and $({\epsilon }_1(x,\beta ),{\epsilon }_2(x,\beta ),\dots )$ 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 $n\to \infty$, 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 $x\in (0,1]$. Furthermore, if $\lambda (\beta )>0$, the extremely irregular set containing points $x\in [0,1]$ whose upper limit of $\frac{-{\log }_{\beta }|I_n(x)|}{n}$ equals to $1+ł(\beta )$ is residual, where $1+ł(\beta )$ is a constant 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 is residual for every $\lambda (\beta )>0$.

arXiv comment13 pages, 5 main results