A two-dimensional univoque set
Tom 212 / 2011
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 $q$.
