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On periodic representations in non-Pisot bases

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Baker, Simon, Masakova, Zuzana, Pelantova, Edita and Vavra, Tomas (2017) On periodic representations in non-Pisot bases. Monatshefte fur Mathematik, 184 (1). pp. 1-19. doi:10.1007/s00605-017-1063-9

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Official URL: https://doi.org/10.1007/s00605-017-1063-9

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Abstract

We study periodic expansions in positional number systems with a base β ∈ C, |β| > 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(β), such that all elements of Q(β) 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 β is a rational number or an algebraic integer with no conjugates of modulus 1. We also consider eventually periodic representations of elements of Q(β) 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(β) admits such a representation then β must be a Pisot number or a Salem number. This result generalises a well known result of Schmidt [22].

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Algebraic fields
Journal or Publication Title: Monatshefte fur Mathematik
Publisher: Springer Wien
ISSN: 0026-9255
Official Date: September 2017
Dates:
DateEvent
September 2017Published
29 May 2017Available
22 May 2017Accepted
Volume: 184
Number: 1
Page Range: pp. 1-19
DOI: 10.1007/s00605-017-1063-9
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Open Access
Funder: České Vysoké Učení Technické v Praze [Czech Technical University]
Grant number: SGS14/205/OHK4/3T/14
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
13-03538SGrantová Agentura České Republikyhttp://dx.doi.org/10.13039/501100001824

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