On Periodic Alternate Base Expansions

Émilie Charlier, Célia Cisternino, Savinien Kreczman

Published 2022-06-03Version 1

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 alternate 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 number or a 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. We also give two applications of these results. First, 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. Second, we obtain a property of Pisot numbers, namely that if $\beta$ is a Pisot number then $\beta\in\mathbb{Q}(\beta^p)$ for all integers $p\ge 1$.