A two-dimensional univoque set

Martijn de Vrie ¹   ; Vilmos Komornik ²

¹ Delft University of Technology Mekelweg 4 2628 CD Delft, the Netherlands
² Département de Mathématique Université de Strasbourg 7 rue René Descartes 67084 Strasbourg Cedex, France

Fundamenta Mathematicae, Tome 212 (2011) no. 2, pp. 175-189

Résumé

Let ${\bf J} \subset \mathbb{R}^2$ 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 $(c_i)=c_1c_2\ldots$ is an expansion of $x$ in base $q$. Let $\bf U$ be the set of couples $(x,q) \in \bf J$ such that $x$ has exactly one expansion in base $q$. In this paper we deduce some topological and combinatorial properties of the set $\bf U$. We characterize the closure of $\bf U$, and we determine its Hausdorff dimension. For $(x,q) \in \bf J$, we also prove new properties of the lexicographically largest expansion of $x$ in base $q$.

Détail

DOI : 10.4064/fm212-2-4

Mots-clés : subset mathbb set couples has least representation form sum infty i integer coefficients satisfying say ldots expansion base set couples has exactly expansion base paper deduce topological combinatorial properties set characterize closure determine its hausdorff dimension prove properties lexicographically largest expansion base