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On a question of Mendès France on normal numbers

On a question of Mendès France on normal numbers

Authors :
Manfred G. Madritsch
Verónica Becher
Universidad de Buenos Aires [Buenos Aires] (UBA)
Institut Élie Cartan de Lorraine (IECL)
Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
The first author is supported by grant STIC-Amsud 20-STIC-06 and PICT-2018-02315.
The second author is supported by project ANR-18-CE40-0018 funded by the French National Research Agency.
ANR-18-CE40-0018,EST,Représentations, systèmes dynamiques et pavages(2018)
Madritsch, Manfred
APPEL À PROJETS GÉNÉRIQUE 2018 - Représentations, systèmes dynamiques et pavages - - EST2018 - ANR-18-CE40-0018 - AAPG2018 - VALID
Source :
Acta Arithmetica. 203:271-288
Publication Year :
2022
Publisher :
Institute of Mathematics, Polish Academy of Sciences, 2022.

Abstract

In 2008 or earlier, Michel Mend\`es France asked for an instance of a real number $x$ such that both $x$ and $1/x$ are simply normal to a given integer base $b$. We give a positive answer to this question by constructing a number $x$ such that both $x$ and its reciprocal $1/x$ are continued fraction normal as well as normal to all integer bases greater than or equal to $2$. Moreover, $x$ and $1/x$ are both computable.<br />Comment: 15 pages. arXiv admin note: text overlap with arXiv:1704.03622

Details

ISSN :
17306264 and 00651036
Volume :
203
Database :
OpenAIRE
Journal :
Acta Arithmetica
Accession number :
edsair.doi.dedup.....6011ffa2a25d96a3789603a24dfc5660
Full Text :
https://doi.org/10.4064/aa210813-28-1