Acta Mathematica Hungarica

, Volume 139, Issue 3, pp 208–227

# Purely periodic expansions in systems with negative base

Article

## 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 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 ($$\mathop {\mathrm {Fin}}\nolimits \, (\beta)= {\mathbb {Z}}[\beta]$$) is not only sufficient, but also necessary in the case of positive quadratic and cubic bases. We show that $$\mathop {\mathrm {Fin}}\nolimits \, (-\beta)= {\mathbb {Z}}[\beta]$$ is not necessary in the case of negative bases.

## Key words and phrases

negative base periodic expansions Pisot numbers

11K16 11A63

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