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Periodic representations in algebraic bases

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Abstract

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 \(\mathcal A\).

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Acknowledgements

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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Authors and Affiliations

  1. Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18600, Prague 8, Czech Republic

    Vítězslav Kala & Tomáš Vávra

  2. Mathematisches Institut, University of Göttingen, Bunsenstr. 3-5, 37073, Göttingen, Germany

    Vítězslav Kala

Authors
  1. Vítězslav Kala
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  2. Tomáš Vávra
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Correspondence to Tomáš Vávra.

Additional information

Communicated by A. Constantin.

The authors were supported by Czech Science Foundation GAČR, Grant 17-04703Y.

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Kala, V., Vávra, T. Periodic representations in algebraic bases. Monatsh Math 188, 109–119 (2019). https://doi.org/10.1007/s00605-017-1151-x

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  • DOI: https://doi.org/10.1007/s00605-017-1151-x

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