On Periodic Alternate Base Expansions

For an alternate base $\boldsymbol{\beta}=(\beta_0,\ldots,\beta_{p-1})$, we show that if all rational numbers in the unit interval $[0,1)$ have periodic expansions with respect to the $p$ shifts of $\boldsymbol{\beta}$, then the bases $\beta_0,\ldots,\beta_{p-1}$ all belong to the extension field $\mathbb Q(\beta)$ where $\beta$ is the product $\beta_0\cdots\beta_{p-1}$ and moreover, this product $\beta$ must be either a Pisot or Salem number. We also prove the stronger statement that if the bases $\beta_0,\ldots,\beta_{p-1}$ belong to $\mathbb Q(\beta)$ but the product $\beta$ is neither a Pisot number nor a Salem number then the set of rationals having an ultimately periodic $\boldsymbol{\beta}$-expansion is nowhere dense in $[0,1)$. Moreover, in the case where the product $\beta$ is a Pisot number and the bases $\beta_0,\ldots,\beta_{p-1}$ all belong to $\mathbb Q(\beta)$, we prove that the set of points in $[0,1)$ having an ultimately periodic $\boldsymbol{\beta}$-expansion is precisely the set $\mathbb Q(\beta)\cap[0,1)$. For the restricted case of R\'enyi 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 $\boldsymbol{\beta}=(\beta_0,\ldots,\beta_{p-1})$ is an alternate base such that the product $\beta$ of the bases is a Pisot number and $\beta_0,\ldots,\beta_{p-1}\in\mathbb Q(\beta)$, then $\boldsymbol{\beta}$ is a Parry alternate base, meaning that the quasi-greedy expansions of $1$ with respect to the $p$ shifts of the base $\boldsymbol{\beta}$ are ultimately periodic.