Typical points of univoque sets

Abstract

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 $\{0,1,\cdots ,M\}$. In this paper we show that for any $x\in {\mathcal{U}}_q$ and all sufficiently small $\epsilon >0$ the Hausdorff dimension ${\dim }_H{\mathcal{U}}_q\cap (x-\epsilon ,x+\epsilon )$ 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).

This paper has been withdrawn by the author due to a crucial error in Theorem 3.5

References in corpus (1)

• Greedy and quasi-greedy expansions in non-integer bases