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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. © Springer International Publishing AG, part of Springer Nature 2018.

Registro:

Documento: Artículo
Título:Normal numbers and computer science
Autor:Becher, V.; Carton, O.
Filiación:Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires. CONICET, Pabellón, Ciudad Universitaria, Buenos Aires, C1428EGA, Argentina
IRIF, UMR 8243, CNRS and Université Paris Diderot, Case 7014, Paris Cedex 13, F-75205, France
Año:2018
Número:9783319691510
Página de inicio:233
Página de fin:269
DOI: http://dx.doi.org/10.1007/978-3-319-69152-7_7
Título revista:Trends in Mathematics
Título revista abreviado:Trends Math.
ISSN:22970215
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_22970215_v_n9783319691510_p233_Becher

Citas:

---------- APA ----------
Becher, V. & Carton, O. (2018) . Normal numbers and computer science. Trends in Mathematics(9783319691510), 233-269.
http://dx.doi.org/10.1007/978-3-319-69152-7_7
---------- CHICAGO ----------
Becher, V., Carton, O. "Normal numbers and computer science" . Trends in Mathematics, no. 9783319691510 (2018) : 233-269.
http://dx.doi.org/10.1007/978-3-319-69152-7_7
---------- MLA ----------
Becher, V., Carton, O. "Normal numbers and computer science" . Trends in Mathematics, no. 9783319691510, 2018, pp. 233-269.
http://dx.doi.org/10.1007/978-3-319-69152-7_7
---------- VANCOUVER ----------
Becher, V., Carton, O. Normal numbers and computer science. Trends Math. 2018(9783319691510):233-269.
http://dx.doi.org/10.1007/978-3-319-69152-7_7