Periodicity and pure periodicity in alternate base systems

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F24%3A00375681" target="_blank" >RIV/68407700:21340/24:00375681 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1007/s40993-024-00542-5" target="_blank" >https://doi.org/10.1007/s40993-024-00542-5</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s40993-024-00542-5" target="_blank" >10.1007/s40993-024-00542-5</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Periodicity and pure periodicity in alternate base systems

Popis výsledku v původním jazyce

We study the Cantor real base numeration system which is a common generalization of two positional systems, namely the Cantor system with a sequence of integer bases and the Rényi system with one real base. We focus on the case of an alternate base B given by a purely periodic sequence of real numbers greater than 1. We answer an open question of Charlier et al. (J Number Theory 254:184–198, 2024) on the set of numbers with eventually periodic B-expansions. We also investigate for which bases all sufficiently small rationals have a purely periodic B-expansion. We show that a necessary condition for this phenomenon is that (where p is the period-length of B) is a Pisot or a Salem unit. We also provide a sufficient condition. We thus generalize the results known for the Rényi numeration system, i.e. for the case when p=1. We provide a class of alternate bases in which all rational numbers in the interval [0,1) have a purely periodic B-expansion.

Název v anglickém jazyce

Periodicity and pure periodicity in alternate base systems

Popis výsledku anglicky

We study the Cantor real base numeration system which is a common generalization of two positional systems, namely the Cantor system with a sequence of integer bases and the Rényi system with one real base. We focus on the case of an alternate base B given by a purely periodic sequence of real numbers greater than 1. We answer an open question of Charlier et al. (J Number Theory 254:184–198, 2024) on the set of numbers with eventually periodic B-expansions. We also investigate for which bases all sufficiently small rationals have a purely periodic B-expansion. We show that a necessary condition for this phenomenon is that (where p is the period-length of B) is a Pisot or a Salem unit. We also provide a sufficient condition. We thus generalize the results known for the Rényi numeration system, i.e. for the case when p=1. We provide a class of alternate bases in which all rational numbers in the interval [0,1) have a purely periodic B-expansion.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10100 - Mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2024

Kód důvěrnosti údajů

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

Název periodika

Research in Number Theory

ISSN

2363-9555

e-ISSN

2363-9555

Svazek periodika

10

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

18

Strana od-do

—

Kód UT WoS článku

001246592200004

EID výsledku v databázi Scopus

2-s2.0-85195483120