Periodicity and pure periodicity in alternate base systems
The result's identifiers
Result code in 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>
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
Periodicity and pure periodicity in alternate base systems
Original language description
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.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10100 - Mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Research in Number Theory
ISSN
2363-9555
e-ISSN
2363-9555
Volume of the periodical
10
Issue of the periodical within the volume
3
Country of publishing house
CH - SWITZERLAND
Number of pages
18
Pages from-to
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UT code for WoS article
001246592200004
EID of the result in the Scopus database
2-s2.0-85195483120