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Normal Numbers and Computer Science

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Sequences, Groups, and Number Theory

Part of the book series: Trends in Mathematics ((TM))

Abstract

Émile Borel defined normality more than 100 years ago to formalize the most basic form of randomness for real numbers. A number is normal to a given integer base if its expansion in that base is such that all blocks of digits of the same length occur in it with the same limiting frequency. This chapter is an introduction to the theory of normal numbers. We present five different equivalent formulations of normality, and we prove their equivalence in full detail. Four of the definitions are combinatorial, and one is, in terms of finite automata, analogous to the characterization of Martin-Löf randomness in terms of Turing machines. All known examples of normal numbers have been obtained by constructions. We show three constructions of numbers that are normal to a given base and two constructions of numbers that are normal to all integer bases. We also prove Agafonov’s theorem that establishes that a number is normal to a given base exactly when its expansion in that base is such that every subsequence selected by a finite automaton is also normal.

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Author information

Authors and Affiliations

  1. Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires. CONICET, Pabellón, Ciudad Universitaria, C1428EGA, Buenos Aires, Argentina

    Verónica Becher

  2. IRIF, UMR 8243, CNRS & Université Paris Diderot, Case 7014, F-75205, Paris Cedex 13, France

    Olivier Carton

Authors
  1. Verónica Becher
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  2. Olivier Carton
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Correspondence to Verónica Becher .

Editor information

Editors and Affiliations

  1. IRIF, Université Paris Diderot, Paris, France

    Valérie Berthé

  2. Department of Mathematics, University of Liège, Liège, Belgium

    Michel Rigo

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Becher, V., Carton, O. (2018). Normal Numbers and Computer Science. In: Berthé, V., Rigo, M. (eds) Sequences, Groups, and Number Theory. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-69152-7_7

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