Beta-integers (``$\beta$-integers'') are those numbers which are the
counterparts of integers when real numbers are expressed in irrational basis
$\beta > 1$. In quasicrystalline studies $\beta$-integers supersede the
``crystallographic'' ordinary integers. When the number $\beta$ is a Parry
number, the corresponding $\beta$-integers realize only a finite number of
distances between consecutive elements and somewhat appear like ordinary
integers, mainly in an asymptotic sense. In this letter we make precise this
asymptotic behavior by proving four theorems concerning Parry $\beta$-integers.Comment: 17 page

Balková, L.

Gazeau, J. P.

Pelantová, E.

English

arXiv

Beta-integers ("β-integers") are those numbers which are the counterparts of integers when real numbers are expressed in an irrational base β > 1. In quasicrystalline studies, β-integers supersede the "crystallographic" ordinary integers. When the number β is a Parry number, the corresponding β-integers realize only a finite number of distances between consecutive elements and are in this sense the most comparable to ordinary integers. In this paper, we point out the similarity of β-integers and ordinary integers in the asymptotic sense, in particular for a subclass of Parry numbers - Pisot numbers for which their Parry and minimal polynomial coincide

Gazeau, J.-P.

Hal-Diderot

Asymptotic Behavior of Beta-Integers

summary: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

Asymptotic behavior of beta-integers

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