On Periodic Alternate Base Expansions

Abstract

For an alternate base β = (β0, . . . , βp−1), we show that if all rational numbers in the unit interval [0, 1) have periodic alternate expansions with respect to the p shifts of β, then the bases β0, . . . , βp−1 all belong to the extension field Q(β) where β is the product β0 • • • βp−1 and moreover, this product β must be either a Pisot number or a Salem number. We also prove the stronger statement that if the bases β0, . . . , βp−1 belong to Q(β) but the product β is neither a Pisot number nor a Salem number then the set of rationals having an ultimately periodic β-expansion is nowhere dense in [0, 1). Moreover, in the case where the product β is a Pisot number and the bases β0, . . . , βp−1 all belong to Q(β), we prove that the set of points in [0, 1) having an ultimately periodic β-expansion is precisely the set Q(β) ∩ [0, 1). For the restricted case of Rényi real bases, i.e., for p = 1 in our setting, our method gives rise to an elementary proof of Schmidt's original result. We also give two applications of these results. First, if β = (β0, . . . , βp−1) is an alternate base such that the product β of the bases is a Pisot number and β0, . . . , βp−1 ∈ Q(β), then β is a Parry alternate base, meaning that the quasi-greedy expansions of 1 with respect to the p shifts of the base β are ultimately periodic. Second, we obtain a property of Pisot numbers, namely that if β is a Pisot number then β ∈ Q(β p ) for all integers p ≥ 1.