A two-dimensional univoque set

Tom 212 / 2011

Martijn de Vrie, Vilmos Komornik Fundamenta Mathematicae 212 (2011), 175-189 MSC: Primary 11A63; Secondary 11B83. DOI: 10.4064/fm212-2-4

Streszczenie

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 $q$.

Autorzy

  • Martijn de VrieDelft University of Technology
    Mekelweg 4
    2628 CD Delft, the Netherlands
    e-mail
  • Vilmos KomornikDépartement de Mathématique
    Université de Strasbourg
    7 rue René Descartes
    67084 Strasbourg Cedex, France
    e-mail

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