Typical points of univoque sets

TL;DR

This paper studies the structure of univoque sets related to unique base-$q$ expansions, revealing that typical points exhibit full Hausdorff dimension and characterizing when all points are typical, extending previous results.

Contribution

It provides a complete description of typical points in univoque sets and shows that the set of such points has full Hausdorff dimension, strengthening earlier findings.

Findings

Typical points have Hausdorff dimension equal to the entire set or zero.

The set of typical points has full Hausdorff dimension.

If the univoque set is a Cantor set, then all points are typical.

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 $\set{0,1,\cdots,M}$. 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).