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arXiv · 1202.1948

Purely periodic expansions in systems with negative base

Zuzana Masáková·Edita Pelantová
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Abstract

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 $\beta$, we find a sufficient condition so that there exist an interval $J$ containing the origin such that the $(-\beta)$-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 (${\rm Fin}(\beta)=\Z[\beta]$) is not only sufficient, but also necessary in the case of positive quadratic and cubic bases. We show that ${\rm Fin}(-\beta)=\Z[\beta]$ is not necessary in the case of negative bases.

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Zuzana Masáková, Edita Pelantová. 2012-02-09. Purely periodic expansions in systems with negative base. https://arxiv.org/abs/1202.1948

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