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On the construction of absolutely normal numbers

Volume 180 / 2017

Christoph Aistleitner, Verónica Becher, Adrian-Maria Scheerer, Theodore A. Slaman Acta Arithmetica 180 (2017), 333-346 MSC: Primary 11K16; Secondary 11Y16, 68-04. DOI: 10.4064/aa170213-5-8 Published online: 28 September 2017

Abstract

We give a construction of an absolutely normal real number $x$ such that for every integer $b \ge 2$, the discrepancy of the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 0}$ is of asymptotic order $\mathcal{O}(N^{-1/2})$. This is below the order of discrepancy which holds for almost all real numbers. Even the existence of absolutely normal numbers having a discrepancy of such a small asymptotic order has not been known before.

Authors

  • Christoph AistleitnerInstitute of Analysis and Number Theory
    Graz University of Technology
    A-8010 Graz, Austria
    e-mail
  • Verónica BecherDepartamento de Computación
    Facultad de Ciencias Exactas y Naturales
    Universidad de Buenos Aires & ICC, CONICET
    Pabellón I, Ciudad Universitaria
    C1428EGA Buenos Aires, Argentina
    e-mail
  • Adrian-Maria ScheererInstitute of Analysis and Number Theory
    Graz University of Technology
    A-8010 Graz, Austria
    e-mail
  • Theodore A. SlamanDepartment of Mathematics
    University of California Berkeley
    719 Evans Hall #3840
    Berkeley, CA 94720-3840, U.S.A.
    e-mail

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