Univoque sets for real numbers

Fan Lü¹ ; Bo Tan¹ ; Jun Wu¹

¹ School of Mathematics and Statistics Huazhong University of Science and Technology 430074, Wuhan, P.R. China

Fundamenta Mathematicae, Tome 227 (2014) no. 1, pp. 69-83

Résumé

For $x\in (0,1)$, the univoque set for $x$, denoted $\mathcal {U}(x)$, is defined to be the set of $\beta \in (1,2)$ such that $x$ has only one representation of the form $x=x_{1}/\beta +x_{2}/\beta ^{2}+\cdots $ with $x_{i}\in \{0,1\}$. We prove that for any $x\in (0,1)$, $\mathcal {U}(x)$ contains a sequence $\{\beta _{k}\}_{k\geq 1}$ increasing to $2$. Moreover, $\mathcal {U}(x)$ is a Lebesgue null set of Hausdorff dimension $1$; both $\mathcal {U}(x)$ and its closure $\overline {\mathcal {U}(x)}$ are nowhere dense.

Détail

DOI : 10.4064/fm227-1-5

Keywords: univoque set denoted mathcal defined set beta has only representation form beta beta cdots prove mathcal contains sequence beta geq increasing moreover mathcal lebesgue null set hausdorff dimension mathcal its closure overline mathcal nowhere dense