Asymptotic behavior of beta-integers

Details Asymptotic behavior of beta-integers Asymptotic behavior of beta-integers

2008-02-21

arxiv.org/abs/0802.2992v1

Physics

Mathematical Physics

17 pages

Scientific paper

10.1007/s11005-008-0241-z

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.

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