A two-dimensional univoque set

TL;DR

This paper investigates the set of real numbers with unique base-q expansions, characterizing its topological structure, closure, Hausdorff dimension, and properties of the largest expansion in base q.

Contribution

It provides a detailed topological and combinatorial analysis of the univoque set and introduces new properties of the lexicographically largest expansions.

Findings

Characterized the closure of the univoque set.

Determined the Hausdorff dimension of the univoque set.

Proved new properties of the lexicographically largest expansion.

Abstract

Let $\mathbf{J} \subset \mathbb{R}^2$ be the set of couples $(x,q)$ with $q>1$ such that $x$ has at least one representation of the form $x=\sum_{i=1}^{\infty} c_i q^{-i}$ with integer coefficients $c_i$ satisfying $0 \leq c_i < q$, $i \ge 1$. In this case we say that $(c_i)=c_1c_2...$ is an expansion of $x$ in base $q$. Let $\mathbf{U}$ be the set of couples $(x,q) \in \mathbf{J}$ such that $x$ has exactly one expansion in base $q$. In this paper we deduce some topological and combinatorial properties of the set $\mathbf{U}$. We characterize the closure of $\mathbf{U}$, and we determine its Hausdorff dimension. For $(x,q) \in \mathbf{J}$, we also prove new properties of the lexicographically largest expansion of $x$ in base $q$.