Arithmetics in numeration systems with negative quadratic base

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.

💡 Research Summary

The paper investigates positional numeration systems with a negative real base, denoted $-\beta$, a concept originally introduced by Ito and Sadahiro. The authors focus on the arithmetic behavior of such systems when the base $\beta$ is a quadratic Pisot number, i.e., a real root greater than one of a quadratic polynomial with integer coefficients whose other root lies inside the unit circle. Specifically, they consider the family of bases defined as the larger root of
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