On a problem of countable expansions

Abstract

For a real number $q\in (1,2)$ and $x\in [0,1/(q-1)]$, the infinite sequence $(d_i)$ is called a \emph{$q$-expansion} of $x$ if $$x=\sum _{i=1}^{\infty }\frac{d_i}{q^i},d_i\in \{0,1\}\text{for all}i\ge 1.$$ For $m=1,2,\cdots$ or ${\aleph }_0$ we denote by ${\mathcal{B}}_m$ the set of $q\in (1,2)$ such that there exists $x\in [0,1/(q-1)]$ having exactly $m$ different $q$-expansions. It was shown by Sidorov (2009) that $q_2:=\min {\mathcal{B}}_2\approx 1.71064$, and later asked by Baker (2015) whether $q_2\in {\mathcal{B}}_{{\aleph }_0}$? In this paper we provide a negative answer to this question and conclude that ${\mathcal{B}}_{{\aleph }_0}$ is not a closed set. In particular, we give a complete description of $x\in [0,1/(q_2-1)]$ having exactly two different $q_2$-expansions.

18 pages, 1 table; To appear in J. Number Theory

References in corpus (2)

• Expansions in non-integer bases: lower, middle and top orders
• Hausdorff dimension of unique beta expansions

Cited by in corpus (5)

• Multiple expansions of real numbers with digits set $\{0,1,q\}$
• Univoque graphs and multiple expansions
• Expansions in multiple bases over general alphabets
• Bases in which some numbers have exactly two expansions
• Smallest bases of expansions with multiple digits