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2 results on '"11K16, 11J70"'

1. On a question of Mend\`es France on normal numbers

Author

Becher, Verónica and Madritsch, Manfred G.

Subjects
Mathematics - Number Theory, 11K16, 11J70
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., Comment: 15 pages. arXiv admin note: text overlap with arXiv:1704.03622

Published
2021

2. On a question of Mendès France on normal numbers

Author

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, and APPEL À PROJETS GÉNÉRIQUE 2018 - Représentations, systèmes dynamiques et pavages - - EST2018 - ANR-18-CE40-0018 - AAPG2018 - VALID

Subjects
Algebra and Number Theory, Mathematics - Number Theory, 11K16, 11J70, MSC2020: 11K16, 11J70, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
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., Comment: 15 pages. arXiv admin note: text overlap with arXiv:1704.03622

Published
2022
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