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

TL;DR

This paper investigates the topological structure of irregular sets in beta-expansions, showing that certain accumulation point sets are always closed intervals and that some irregular sets are residual, revealing complex fractal properties.

Contribution

It proves that the accumulation point set of the normalized negative log-lengths in beta-expansions is always a closed interval and characterizes the residuality of extremely irregular sets.

Findings

The set of accumulation points is always a closed interval.

Certain irregular sets are residual when a specific measure is positive.

The irregular set with non-existent limits is residual for all positive measure cases.

Abstract

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 $|I_n(x)|$ 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 $1+\l(\beta)$ is residual, where $1+\l(\beta)$ 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 $\lambda(\beta)>0$.