On periodic representations in non-Pisot bases

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

On periodic representations in non-Pisot bases

Popis výsledku v původním jazyce

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).

Název v anglickém jazyce

On periodic representations in non-Pisot bases

Popis výsledku anglicky

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).

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA13-03538S" target="_blank" >GA13-03538S: Algoritmy, dynamika a geometrie numeračních systémů</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2017

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Monatshefte für Mathematik

ISSN

0026-9255

e-ISSN

1436-5081

Svazek periodika

184

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

AT - Rakouská republika

Počet stran výsledku

19

Strana od-do

1-19

Kód UT WoS článku

000407394400001

EID výsledku v databázi Scopus

2-s2.0-85019673306