Arithmetics in numeration systems with negative quadratic base

TL;DR

This paper investigates the arithmetical properties of negative quadratic base numeration systems, especially for quadratic Pisot numbers, revealing closure properties and bounds on fractional digits in such systems.

Contribution

It characterizes the set of finite expansions in negative quadratic bases, showing closure under addition and identifying bounds on fractional digits for specific bases.

Findings

Finite $(-eta)$-expansions are closed under addition.

Bound on fractional digits for $eta= au$ (golden ratio) is determined.

Set of $(- au)$-integers coincides with $( au^2)$-integers on positive half-line.

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 $x^2-mx-n$, $m\geq n\geq 1$, and show that in this case the set ${\rm Fin}(-\beta)$ of finite $(-\beta)$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $\beta=\tau=\frac12(1+\sqrt5)$, 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.