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arXiv · 1011.1403

Arithmetics in numeration systems with negative quadratic base

Z. Masáková·T. Vávra
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Abstract

We consider positional numeration system with negative base $-β$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $β$ is a quadratic Pisot number. We study a class of roots $β>1$ of polynomials $x^2-mx-n$, $m\geq n\geq 1$, and show that in this case the set ${\rm Fin}(-β)$ of finite $(-β)$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $β=τ=\frac12(1+\sqrt5)$, the golden ratio. For such $β$, we determine the exact bound on the number of fractional digits appearing in arithmetical operations. We also show that the set of $(-τ)$-integers coincides on the positive half-line with the set of $(τ^2)$-integers.

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Z. Masáková, T. Vávra. 2010-11-05. Arithmetics in numeration systems with negative quadratic base. https://arxiv.org/abs/1011.1403

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