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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
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 | ||||||||
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| 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: |
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| 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: |
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