On periodic representations in non-Pisot bases

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F17%3A00305566" target="_blank" >RIV/68407700:21340/17:00305566 - isvavai.cz</a>
Result on the web
<a href="http://link.springer.com/article/10.1007/s00605-017-1063-9" target="_blank" >http://link.springer.com/article/10.1007/s00605-017-1063-9</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00605-017-1063-9" target="_blank" >10.1007/s00605-017-1063-9</a>

Result language
angličtina
Original language name
On periodic representations in non-Pisot bases
Original language description
We study periodic expansions in positional number systems with a base βelementC, |β|>1, and with coefficients in a finite set of digits AcC. We are interested in determining those algebraic bases for which there exists AcQ(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 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 (Bull Lond Math Soc 12(4):269–278, 1980).
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GA13-03538S" target="_blank" >GA13-03538S: Algorithms, Dynamics and Geometry of Numeration systems</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2017
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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