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For an alternate base $\boldsymbol{\beta}=(\beta_0,\ldots,\beta_{p-1})$, weshow that if all rational numbers in the unit interval $[0,1)$ have periodicexpansions with respect to the $p$ shifts of $\boldsymbol{\beta}$, then thebases $\beta_0,\ldots,\beta_{p-1}$ all belong to the extension field $\mathbbQ(\beta)$ 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 thestronger 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 aSalem number then the set of rationals having an ultimately periodic$\boldsymbol{\beta}$-expansion is nowhere dense in $[0,1)$. Moreover, in thecase 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 thatthe set of points in $[0,1)$ having an ultimately periodic$\boldsymbol{\beta}$-expansion is precisely the set $\mathbbQ(\beta)\cap[0,1)$. 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'soriginal result. Therefore, even though our results generalize those ofSchmidt, our proofs should not be seen as generalizations of Schmidt's originalarguments but as an original method in the generalized framework of alternatebases, which moreover gives a new elementary proof of Schmidt's results from1980. As an application of our results, we show that if$\boldsymbol{\beta}=(\beta_0,\ldots,\beta_{p-1})$ is an alternate base suchthat 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 aParry alternate base, meaning that the quasi-greedy expansions of $1$ withrespect to the $p$ shifts of the base $\boldsymbol{\beta}$ are ultimatelyperiodic.

On Periodic Alternate Base Expansions