On periodic alternate base expansions

For an alternate base B=(β_0,...,β_{p-1}), we show that if all rational numbers in the unit interval [0, 1) have periodic expansions with respect to all shifts of B, then the bases β_0,...,β_{p-1} all belong to the extension field Q(β) where β is the product
β_0 · · · β_{p−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 β_0, ..., β_{p−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 β_0, ..., β_{p−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).