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{á}bor Kall{ó}s (1999, 2001). Moreover, we find that the dimension function $f(\beta )={\dim }_H{\mathbf{U}}_{\beta ,N}$ fluctuates frequently for $\beta \in (1,N)$.

28 pages,4 figures

References in corpus (2)

• Greedy and quasi-greedy expansions in non-integer bases
• A two-dimensional univoque set

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• On small univoque bases of real numbers
• Critical base for the unique codings of fat Sierpinski gasket
• Univoque graphs and multiple expansions
• Hausdorff distance of univoque sets
• Density spectrum of Cantor measure
• Digit frequencies of beta-expansions
• Difference of Cantor sets and frequencies in Thue--Morse type sequences