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Topic: [HM] question about term "normal"
Replies: 1   Last Post: Oct 28, 2006 4:01 PM

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Udai Venedem

Posts: 19
Registered: 2/16/05
Re: [HM] question about term "normal"
Posted: Oct 28, 2006 4:01 PM
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Dear all Borelians,
indeed, Borel is the one who coined the term and developed the concept
of "normal" about the representation of the numbers in a given base.
Here are some precisions and references.

Peter Flor:
> in the book "Uniform dustribution of sequences" by KUIPERS and
> NIEDERREITER.
> According to these authors, the concept dates back to some publications
> by Emile Borel, of 1909 and 1914.


1914 "most probably" refers to:
- LE HASARD. Paris, Alcan 1914. Small in-8

1909 is, of course the famous milestone paper:
- Les probabilités dénombrables et leur applications arithmétiques, p.
247-271 in RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO. T. 27 (1°
semestre 1909), Palerme 1909. Great in-8.
where indeed, on p. 260, he defines "simplement normal" (in base 10) a
fractional number when the frequency of its figures is 1/10. Then he
goes on refining the concept, and defines "entièrement normal", and
"absolument normal" numbers (this last refinement is on p. 261).

I find another important reference in 1926:
- APPLICATIONS A L'ARITHMÉTIQUE ET A LA THÉORIE DES FONCTIONS Leçons
professées à la Faculté des Sciences de Paris rédigées par Paul
DUBREIL. Paris, Gauthier-Villars, 1926. Great in-8. (Premier fascicule
du Tome II du Traité du calcul des probabilités et de ses applications).
where the first chapter goes back to the definitions quoted supra.

Udai Venedem
http://aaaa.fr.eu.org/alta.mathematica/
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