A two-dimensional univoque set

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 $0\le c_i<q$, $i\ge 1$. In this case we say that $(c_i)=c_1{c}_2...$ 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 $q$.

12 pages, 0 figures. Comments of the referee are incorporated. Accepted for publication in Fundamenta Mathematicae

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