On small bases which admit countably many expansions

TL;DR

This paper investigates the set of bases q in (1,2) for which some number x has exactly countably many q-expansions, identifying the smallest such base above the golden ratio and exploring the structure of these expansions.

Contribution

It determines the minimal base greater than the golden ratio with countably many expansions and establishes a dichotomy in the number of expansions for bases in a specific interval.

Findings

The smallest base greater than the golden ratio with countably many expansions is approximately 1.64541.

A full dichotomy for the number of q-expansions in the interval (golden ratio, q_{aleph_0}) is established.

If a number has uncountably many q-expansions, then the set of expansions has the cardinality of the continuum.

Abstract

Let $q\in(1,2)$ and $x\in[0,\frac1{q-1}]$. We say that a sequence $(\epsilon_i)_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ is an expansion of $x$ in base $q$ (or a $q$-expansion) if x=\sum_{i=1}^{\infty}\epsilon_iq^{-i}. Let $\mathcal{B}_{\aleph_{0}}$ denote the set of $q$ for which there exists $x$ with exactly $\aleph_{0}$ expansions in base $q$. In \cite{EHJ} it was shown that $\min\mathcal{B}_{\aleph_{0}}=\frac{1+\sqrt{5}}{2}.$ In this paper we show that the smallest element of $\mathcal{B}_{\aleph_{0}}$ strictly greater than $\frac{1+\sqrt{5}}{2}$ is $q_{\aleph_{0}}\approx1.64541$, the appropriate root of $x^6=x^4+x^3+2x^2+x+1$. This leads to a full dichotomy for the number of possible $q$-expansions for $q\in (\frac{1+\sqrt{5}}{2},q_{\aleph_{0}})$. We also prove some general results regarding $\mathcal{B}_{\aleph_{0}}\cap[\frac{1+\sqrt{5}}{2},q_{f}],$ where $q_{f}\approx 1.75488$ is the appropriate root of $x^{3}=2x^{2}-x+1.$ Moreover, the techniques developed in this paper imply that if $x\in [0,\frac{1}{q-1}]$ has uncountably many $q$-expansions then the set of $q$-expansions for $x$ has cardinality equal to that of the continuum, this proves that the continuum hypothesis holds when restricted to this specific case.