Hausdorff dimension of unique beta expansions

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 ${\beta}\in(1,N)$.