Bases in which some numbers have exactly two expansions

Abstract

In this paper we answer several questions raised by Sidorov on the set ${\mathcal{B}}_2$ of bases in which there exist numbers with exactly two expansions. In particular, we prove that the set ${\mathcal{B}}_2$ is closed, and it contains both infinitely many isolated and accumulation points in $(1,q_{KL})$, where $q_{KL}\approx 1.78723$ is the Komornik-Loreti constant. Consequently we show that the second smallest element of ${\mathcal{B}}_2$ is the smallest accumulation point of ${\mathcal{B}}_2$. We also investigate the higher order derived sets of ${\mathcal{B}}_2$. Finally, we prove that there exists a $\delta >0$ such that \begin{equation*} \dim_H(\mathcal B_2\cap(q_{KL}, q_{KL}+δ))<1, \end{equation*} where ${\dim }_H$ denotes the Hausdorff dimension.

34 pages, 1 figure. To appear in J. Number Theory

References in corpus (3)

• Expansions in non-integer bases: lower, middle and top orders
• Greedy and quasi-greedy expansions in non-integer bases
• On a problem of countable expansions