On small univoque bases of real numbers

Derong Kong

doi:10.1007/s10474-016-0637-7

paper

On small univoque bases of real numbers

arXiv:1602.06173 · doi:10.1007/s10474-016-0637-7

Abstract

Given a positive real number $x$ , we consider the smallest base $q_{s} (x) \in (1, 2)$ for which there exists a unique sequence $(d_{i})$ of zeros and ones such that \[ x=\sum_{i=1}^\infty\frac{d_i}{(q_s(x))^i}. \] In this paper we give complete characterizations of those $x$ 's for which $q_{s} (x) \leq q_{K L}$ , where $q_{K L}$ is the Komornik-Loreti constant. Furthermore, we show that $q_{s} (x) = q_{K L}$ if and only if \[ x\in\left\{1, ~\frac{q_{KL}}{q_{KL}^2-1},~ \frac{1}{q_{KL}^2-1}, ~\frac{1}{q_{KL}(q_{KL}^2-1)}\right\}. \] Finally, we determine the explicit value of $q_{s} (x)$ if $q_{s} (x) < q_{K L}$ .

16 pages, 1 figure. To appear in Acta Math. Hungar

References in corpus (1)

Hausdorff dimension of unique beta expansions