Periodic representations in algebraic bases
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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 \(\mathcal A\).
KeywordsPisot number Salem number Expansion in non-integer base Periodic representation
Mathematics Subject Classification11A63 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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