Year: 2010
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Masáková, Z., Vávra, T.
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1743211645