Arithmetics in numeration systems with negative quadratic base

Abstract

We consider positional numeration system with negative base $-\beta$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $\beta$ is a quadratic Pisot number. We study a class of roots $\beta >1$ of polynomials $x^2-mx-n$, $m\ge n\ge 1$, and show that in this case the set $\mathrm{Fin}(-\beta )$ of finite $(-\beta )$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $\beta =\tau =\frac{1}{2}(1+\sqrt{5})$, the golden ratio. For such $\beta$, we determine the exact bound on the number of fractional digits appearing in arithmetical operations. We also show that the set of $(-\tau )$-integers coincides on the positive half-line with the set of $({\tau }^2)$-integers.

References in corpus (1)

• Asymptotic behavior of beta-integers