On small bases which admit countably many expansions

الأصل · EN

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) if x=∑ᵢ₌₁∞εᵢq⁻ⁱ. Let B₀ denote the set of q for which there exists x with exactly ₀ expansions in base q. In EHJ it was shown that ₀=1+√52. In this paper we show that the smallest element of B₀ strictly greater than 1+√52 is q₀≈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,q₀). We also prove some general results regarding B₀∩[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∈ [0,1/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.