Letters in Mathematical Physics

, Volume 84, Issue 2–3, pp 179–198 | Cite as

Asymptotic Behavior of Beta-Integers

  • Lubomira Balková
  • Jean-Pierre Gazeau
  • Edita Pelantová


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.


beta-integers beta-numeration asymptotic behavior quasicrystals aperiodic structure 

Mathematics Subject Classification (2000)

11R06 52C23 11B05 


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Copyright information

© Springer 2008

Authors and Affiliations

  • Lubomira Balková
    • 1
    • 2
  • Jean-Pierre Gazeau
    • 2
  • Edita Pelantová
    • 1
  1. 1.Department of MathematicsFNSPE, Czech Technical UniversityPraha 2Czech Republic
  2. 2.Laboratoire APCUniversité Paris 7-Denis DiderotParis Cedex 13France

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