Abstract
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.
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Balková, L., Gazeau, JP. & Pelantová, E. Asymptotic Behavior of Beta-Integers. Lett Math Phys 84, 179–198 (2008). https://doi.org/10.1007/s11005-008-0241-z
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DOI: https://doi.org/10.1007/s11005-008-0241-z