Univoque bases of real numbers: local dimension, Devil's staircase and isolated points

Abstract

Given a positive integer $M$ and a real number $x>0$, let $\mathcal{U}(x)$ be the set of all bases $q\in (1,M+1]$ for which there exists a unique sequence $(d_i)=d_1{d}_2\dots$ with each digit $d_i\in \{0,1,\dots ,M\}$ satisfying $$x=\sum _{i=1}^{\infty }\frac{d_i}{q^i}.$$ The sequence $(d_i)$ is called a $q$-expansion of $x$. In this paper we investigate the local dimension of $\mathcal{U}(x)$ and prove a `variation principle' for unique non-integer base expansions. We also determine the critical values of $\mathcal{U}(x)$ such that when $x$ passes the first critical value the set $\mathcal{U}(x)$ changes from a set with positive Hausdorff dimension to a countable set, and when $x$ passes the second critical value the set $\mathcal{U}(x)$ changes from an infinite set to a singleton. Denote by $\mathbf{U}(x)$ the set of all unique $q$-expansions of $x$ for $q\in \mathcal{U}(x)$. We give the Hausdorff dimension of $\mathbf{U}(x)$ and show that the dimensional function $x\mapsto {\dim }_H\mathbf{U}(x)$ is a non-increasing Devil's staircase. Finally, we investigate the topological structure of $\mathcal{U}(x)$. In contrast with $x=1$ that $\mathcal{U}(1)$ has no isolated points, we prove that for typical $x>0$ the set $\mathcal{U}(x)$ contains isolated points.

26 pages, 2 figures. In this version we simplified the proof of Theorem 1.1