Arithmetics in number systems with negative base

Abstract

We study the numeration system with negative basis, introduced by Ito and Sadahiro. We focus on arithmetic operations in the set $\mathrm{Fin}(-\beta )$ and ${\mathbb{Z}}_{-\beta }$ of numbers having finite resp. integer $(-\beta )$-expansions. We show that $\mathrm{Fin}(-\beta )$ is trivial if $\beta$ is smaller than the golden ratio $\frac{1}{2}(1+\sqrt{5})$. For $\beta \ge \frac{1}{2}(1+\sqrt{5})$ we prove that $\mathrm{Fin}(-\beta )$ is a ring, only if $\beta$ is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that $\mathrm{Fin}(-\beta )$ is a ring if $\beta$ is a quadratic Pisot number with positive conjugate. For quadratic Pisot units we determine the number of fractional digits that may appear when adding or multiplying two $(-\beta )$-integers.

13 pages

Cited by in corpus (1)

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