Hausdorff dimension of unique beta expansions

Given an integer $N\ge 2$ and a real number $β>1$, let $Γ_{β,N}$ be the set of all $x=\sum_{i=1}^\infty {d_i}/{β^i}$ with $d_i\in\{0,1,\cdots,N-1\}$ for all $i\ge 1$. The infinite sequence $(d_i)$ is called a $β$-expansion of $x$. Let $\mathbf{U}_{β,N}$ be the set of all $x$'s in $Γ_{β,N}$ which have unique $β$-expansions. We give explicit formula of the Hausdorff dimension of $\mathbf{U}_{β,N}$ for $β$ in any admissible interval $[{β}_L,{β}_U]$, where ${β_L}$ is a purely Parry number while ${β_U}$ 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 $β>1$. In particular, this improves the main results of G{á}bor Kall{ó}s (1999, 2001). Moreover, we find that the dimension function $f(β)=\dim_H\mathbf{U}_{β,N}$ fluctuates frequently for $β\in(1,N)$.

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