On small bases which admit countably many expansions

Abstract

Let q∈(1,2) and x∈[0,1q−1]. We say that a sequence (ϵi)i=1∞∈{0,1}N is an expansion of x in base q (or a q-expansion) ifx=∑i=1∞ϵiq−i. Let Bℵ0 denote the set of q for which there exists x with exactly ℵ0 expansions in base q. In [5] it was shown that min⁡Bℵ0=1+52. In this paper we show that the smallest element of Bℵ0 strictly greater than 1+52 is qℵ0≈1.64541, the appropriate root of x6=x4+x3+2x2+x+1. This leads to a full dichotomy for the number of possible q-expansions for q∈(1+52,qℵ0). We also prove some general results regarding Bℵ0∩[1+52,qf], where qf≈1.75488 is the appropriate root of x3=2x2−x+1. Moreover, the techniques developed in this paper imply that if x∈[0,1q−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.