Typical points of univoque sets

Given a positive integer $M$ and a real number $q>1$, we consider the univoque set $\mathcal{U}_q$ of reals which have a unique $q$-expansion over the alphabet $\set{0,1,\cdots,M}$. In this paper we show that for any $x\in\mathcal{U}_q$ and all sufficiently small $\varepsilon>0$ the Hausdorff dimension $\dim_H\mathcal{U}_q\cap(x-\varepsilon, x+\varepsilon)$ equals either $\dim_H\mathcal{U}_q$ {or} zero. Moreover, we give a complete description of the typical points $x\in\mathcal{U}_q$ which satisfy $\dim_H\mathcal{U}_q\cap(x-\varepsilon, x+\varepsilon)=\dim_H\mathcal{U}_q\quad\textrm{for any}\quad \varepsilon>0,$ and prove that the set of typical points of $\mathcal{U}_q$ has full Hausdorff dimension. In particular, we show that if $\mathcal{U}_q$ is a Cantor set, then all points of $\mathcal{U}_q$ are typical points. This strengthen a result of de Vries and Komornik (Adv. Math., 2009).

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