负基系统中的周期展开式
Purely periodic expansions in systems with negative base
摘要：
研究负基数制中展开式的纯周期性问题。与Akiyama关于正Pisot单位基$β$的结果类似，我们找到了一个充分条件，使得存在一个包含原点的区间$J$，使得从$J$开始的每个有理数的$(-β)$ -展开式是纯周期的.我们专注于二次基的情况下，并证明了以下的差异之间的负和正的基础：它是已知的有限性性质（${\rm Fin}(β)=\Z[β]$）不仅是充分的，但也是必要的情况下，积极的二次和三次基地。我们表明，${\rm Fin}(-β)=\Z[β]$是不必要的情况下，负基地。
英文摘要：
We study the question of pure periodicity of expansions in the negative base numeration system. In analogy of Akiyama's result for positive Pisot unit base $β$, we find a sufficient condition so that there exist an interval $J$ containing the origin such that the $(-β)$-expansion of every rational number from $J$ is purely periodic. We focus on the case of quadratic bases and demonstrate the following difference between the negative and positive bases: It is known that the finiteness property (${\rm Fin}(β)=\Z[β]$) is not only sufficient, but also necessary in the case of positive quadratic and cubic bases. We show that ${\rm Fin}(-β)=\Z[β]$ is not necessary in the case of negative bases.
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