Unique expansions of real numbers

TL;DR

This paper studies the topological structure of sets of real numbers with unique q-expansions, characterizing when these sets are closed, Cantor sets, or subshifts, revealing complex connections with fractals and measure theory.

Contribution

It provides a detailed topological analysis of univoque sets for each base q, including their closures, conditions for being Cantor sets, and the structure of associated sequence sets.

Findings

Characterized when alU_q is closed or a Cantor set.

Identified bases q for which alU_q is a subshift or of finite type.

Determined when the sequence set alU_q' is constant or has specific symbolic dynamics.

Abstract

It was discovered some years ago that there exist non-integer real numbers $q>1$ for which only one sequence $(c_i)$ of integers $c_i \in [0,q)$ satisfies the equality $\sum_{i=1}^\infty c_iq^{-i}=1$. The set of such "univoque numbers" has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation. In this paper we consider for each fixed $q>1$ the set $\mathcal{U}_q$ of real numbers $x$ having a unique representation of the form $\sum_{i=1}^\infty c_iq^{-i}=x$ with integers $c_i$ belonging to $[0,q)$. We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases $q$ for which $\mathcal{U}_q$ is closed or even a Cantor set. We also study the set $\mathcal{U}_q'$ consisting of all sequences $(c_i)$ of integers $c_i \in [0,q)$ such that $\sum_{i=1}^{\infty} c_i q^{-i} \in \mathcal{U}_q$. We determine the numbers $r >1$ for which the map $q \mapsto \mathcal{U}_q'$ (defined on $(1, \infty)$) is constant in a neighborhood of $r$ and the numbers $q >1$ for which $\mathcal{U}_q'$ is a subshift or a subshift of finite type.