论文标题
在周期性的替代基础扩展
On Periodic Alternate Base Expansions
论文作者
论文摘要
对于替代基础$ \boldsymbolβ=（β_0，\ ldots，β_{p-1}）$，我们表明，如果单位间隔$ [0,1）$中的所有理性数字在$ \boldsymbolβ$中都具有定期扩展$ \ mathbb q（β）$，其中$β$是产品$β_0\cdotsβ_{p-1} $，此外，此产品$β$必须是PISOT或SALEM数字。 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 $[0,1)$.此外，如果产品$β$是PISOT数字，而基础$β_0，\ ldots，β_{p-1} $全部属于$ \ m athbb q（β）$，我们证明了$ [0,1）$的点集，该点具有最终的定期$ \boldsymbolβ$ - expansely $ $blodsymbolβ$ - $blodsymbolβ$ - q（β）\ cap [0,1）$。对于Rényi真实基础的受限案例，即，对于我们的环境中的$ P = 1 $，我们的方法产生了Schmidt原始结果的基本证明。因此，即使我们的结果概括了施密特的结果，我们的证据也不应被视为施密特原始论点的概括，而应在替代基地的一般框架中作为一种原始方法，此外，这给出了施密特的新基本证明，证明了施密特的1980年结果。作为我们的结果的应用，我们表明，如果$ \bloldsymbolβ= $ \bloldsymbolβ= p p p ld.ldots p \ l。 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.
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.
