On Periodic Alternate Base Expansions

Abstract

For an alternate base $\boldsymbolβ=(β_0,\ldots,β_{p-1})$, we show that if all rational numbers in the unit interval $[0,1)$ have periodic expansions with respect to the $p$ shifts of $\boldsymbolβ$, then the bases $β_0,\ldots,β_{p-1}$ all belong to the extension field $\mathbb Q(β)$ where $β$ is the product $β_0\cdotsβ_{p-1}$ and moreover, this product $β$ must be either a Pisot or Salem number. We also prove the stronger statement that if the bases $β_0,\ldots,β_{p-1}$ belong to $\mathbb Q(β)$ but the product $β$ is neither a Pisot number nor a Salem number then the set of rationals having an ultimately periodic $\boldsymbolβ$-expansion is nowhere dense in $[0,1)$. Moreover, in the case where the product $β$ is a Pisot number and the bases $β_0,\ldots,β_{p-1}$ all belong to $\mathbb Q(β)$, we prove that the set of points in $[0,1)$ having an ultimately periodic $\boldsymbolβ$-expansion is precisely the set $\mathbb Q(β)\cap[0,1)$. 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. Therefore, even though our results generalize those of Schmidt, our proofs should not be seen as generalizations of Schmidt's original arguments but as an original method in the generalized framework of alternate bases, which moreover gives a new elementary proof of Schmidt's results from 1980. As an application of our results, we show that if $\boldsymbolβ=(β_0,\ldots,β_{p-1})$ is an alternate base such that the product $β$ of the bases is a Pisot number and $β_0,\ldots,β_{p-1}\in\mathbb Q(β)$, then $\boldsymbolβ$ is a Parry alternate base, meaning that the quasi-greedy expansions of $1$ with respect to the $p$ shifts of the base $\boldsymbolβ$ are ultimately periodic.