Hausdorff dimension of unique beta expansions

TL;DR

This paper derives explicit formulas for the Hausdorff dimension of the set of numbers with unique beta-expansions within certain intervals, revealing complex fluctuation patterns of the dimension function for various beta values.

Contribution

It provides a new explicit formula for the Hausdorff dimension of unique beta-expansions for any beta in admissible intervals, extending previous results and connecting to classical sequences.

Findings

Explicit dimension formulas for beta in admissible intervals.

Dimension function fluctuates frequently for beta in (1,N).

Calculates dimension for almost every beta > 1.

Abstract

Given an integer $N\ge 2$ and a real number ${\beta}>1$, let $\Gamma_{{\beta},N}$ be the set of all $x=\sum_{i=1}^\infty {d_i}/{{\beta}^i}$ with $d_i\in\{0,1,\cdots,N-1\}$ for all $i\ge 1$. The infinite sequence $(d_i)$ is called a ${\beta}$-expansion of $x$. Let $\mathbf{U}_{{\beta},N}$ be the set of all $x$'s in $\Gamma_{{\beta},N}$ which have unique ${\beta}$-expansions. We give explicit formula of the Hausdorff dimension of $\mathbf{U}_{{\beta},N}$ for ${\beta}$ in any admissible interval $[{{\beta}}_L,{{\beta}}_U]$, where ${{\beta}_L}$ is a purely Parry number while ${{\beta}_U}$ is a transcendental number whose quasi-greedy expansion of $1$ is related to the classical Thue-Morse sequence. This allows us to calculate the Hausdorff dimension of $\U{N}$ for almost every $\beta>1$. In particular, this improves the main results of G{\'a}bor Kall{\'o}s (1999, 2001). Moreover, we find that the dimension function $f({\beta})=\dim_H\mathbf{U}_{{\beta},N}$ fluctuates frequently for ${\beta}\in(1,N)$.