Purely periodic expansions in systems with negative base

TL;DR

This paper investigates the conditions under which rational numbers have purely periodic expansions in negative base systems, highlighting differences from positive bases especially in quadratic cases.

Contribution

It establishes a sufficient condition for pure periodicity in negative bases and shows that finiteness property equivalence does not hold for negative bases.

Findings

Existence of an interval where all rationals have purely periodic expansions in negative bases.

Difference between negative and positive bases regarding the finiteness property.

Negative bases can have purely periodic expansions without the finiteness property.

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.