Univoque sets for real numbers
Volume 227 / 2014
Fundamenta Mathematicae 227 (2014), 69-83
MSC: Primary 11K55.
DOI: 10.4064/fm227-1-5
Abstract
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.
