On periodic representations in non-Pisot bases

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We study periodic expansions in positional number systems with a base beta ? C, |beta| > 1, and with coefficients in a finite set of digits A ? C. We are interested in determining those algebraic bases for which there exists A ? Q(beta), such that all elements of Q(beta) admit at least one eventually periodic representation with digits in A. In this paper we prove a general result that guarantees the existence of such an A. This result implies the existence of such an A when beta is a rational number or an algebraic integer with no conjugates of modulus 1. We also consider eventually periodic representations of elements of Q(beta) for which the maximal power of the representation is proportional to the absolute value of the represented number, up to some universal constant. We prove that if every element of Q(beta) admits such a representation then beta must be a Pisot number or a Salem number. This result generalises a well known result of Schmidt [22].


Original languageEnglish
Number of pages19
JournalMonatshefte fur Mathematik
Issue number1
Early online date29 May 2017
Publication statusPublished - 1 Sep 2017


  • Pisot numbers, Expansions in non-integer bases, Periodic representations