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 ${\rm Fin}(-β)$ and $\Z_{-β}$ of numbers having finite resp. integer $(-β)$-expansions. We show that ${\rm Fin}(-β)$ is trivial if $β$ is smaller than the golden ratio $\frac12(1+\sqrt5)$. For $β\geq\frac12(1+\sqrt5)$ we prove that ${\rm Fin}(-β)$ is a ring, only if $β$ is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that ${\rm Fin}(-β)$ is a ring if $β$ 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 $(-β)$-integers.