PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

On a question of Mendès France on normal numbers

Volume 203 / 2022

Verónica Becher, Manfred G. Madritsch Acta Arithmetica 203 (2022), 271-288 MSC: Primary 11K16, 11J70. DOI: 10.4064/aa210813-28-1 Published online: 15 April 2022

Abstract

In 2008 or earlier, Michel Mendès 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 computable, the first $n$ digits of their continued fraction expansion can be obtained in $\mathcal {O}(n^4)$ mathematical operations.

Authors

  • Verónica BecherDepartamento de Computación
    Facultad de Ciencias Exactas y Naturales
    Universidad de Buenos Aires & ICC CONICET
    Pabellón I, Ciudad Universitaria
    1428 Buenos Aires, Argentina
    e-mail
  • Manfred G. MadritschUniversité de Lorraine
    Institut Élie Cartan de Lorraine
    UMR 7502
    F-54506 Vandœuvre-lès-Nancy, France
    and
    CNRS, Institut Élie Cartan de Lorraine
    UMR 7502
    F-54506 Vandœuvre-lès-Nancy, France
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image