On a problem of countable expansions

TL;DR

This paper investigates the structure of real numbers with a specific number of expansions in a non-integer base, proving that certain sets of numbers with countably many expansions are not closed and characterizing those with exactly two expansions.

Contribution

It provides a negative answer to whether the set of numbers with countably many expansions is closed and offers a complete description of numbers with exactly two expansions at a critical base.

Findings

The set of numbers with countably many expansions is not closed.

Complete characterization of numbers with exactly two expansions at the base q_2.

Disproved Baker's question regarding the inclusion of q_2 in the set of numbers with countably many 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 $$ x=\sum_{i=1}^\infty\frac{d_i}{q^i},\quad d_i\in\{0,1\}\quad\textrm{for all}~ i\ge 1. $$ 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 $q_2:=\min \mathcal{B}_2\approx1.71064$, 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.