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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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