Monatshefte für Mathematik

, Volume 188, Issue 1, pp 109–119 | Cite as

Periodic representations in algebraic bases

  • Vítězslav Kala
  • Tomáš VávraEmail author


We study periodic representations in number systems with an algebraic base \(\beta \) (not a rational integer). We show that if \(\beta \) has no Galois conjugate on the unit circle, then there exists a finite integer alphabet \(\mathcal A\) such that every element of \(\mathbb Q(\beta )\) admits an eventually periodic representation with base \(\beta \) and digits in \(\mathcal A\).


Pisot number Salem number Expansion in non-integer base Periodic representation 

Mathematics Subject Classification

11A63 11K16 11R04 



We wish to thank Zuzana Masáková for a careful reading of a draft of the paper and for a number of helpful suggestions.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Algebra, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic
  2. 2.Mathematisches InstitutUniversity of GöttingenGöttingenGermany

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