Nombres normaux applications aux fonctions pseudo-aléatoires

France, Michel Mendès

doi:10.1007/bf02786669

Cited by 26 publications

(17 citation statements)

References 19 publications

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“…More recently, in a paper in this Monthly, Goodman [1] has shown how Kac's approach may be extended so as to obtain deeper results relating to normal numbers, including some of those obtained by Mendès France [4] using more difficult concepts and techniques. The approach of Kac is elementary, and is quite accesible to undergraduate students, except at one point, where it is necessary to invoke the Beppo Levi Theorem to interchange the order of summation and integration in a series of non-negative functions.…”

Section: The Following Results Was Proved Byémile Borel In 1904mentioning

confidence: 99%

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Normal Numbers without Measure Theory

Nillsen¹

2000

The American Mathematical Monthly

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Any number can be expanded to the base 10, leading to a sequence of digits between 0 and 9 corresponding to the number. Also, any number can be expanded to the base 2, leading to a sequence of digits, each one being either 0 or 1, corresponding to the number. It is result due to Émile Borel in 1904 that "almost all" numbers have the property that, when expanded to the base 2, each of the digits 0 and 1 appears with an asymptotic frequency of 1/2. That is, if we regard the sequence of digits in the expansion to the base 2 as a sequence of 'heads' and 'tails' resulting from a coin-tossing experiment, then, in the language of probability theory, the probability of getting heads (that is a 0) is 1/2, and the probability of getting tails (that is a 1) is also 1/2. Numbers with this property are called "simply normal numbers" to the base 2. Traditionally, the proof of Borel's Theorem relies on a knowledge of measure theory, which generally lies outside the undergraduate curriculum. Here, a proof of Borel's Theorem is presented which requires only an introductory knowledge of sequences and series, and a knowledge of how to integrate step functions on an interval. This makes it possible to discuss Borel's theorem at the level of a first or second year course in mathematical analysis.

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Section: The Following Results Was Proved Byémile Borel In 1904mentioning

confidence: 99%

“…It was Mendès France [4] who made a connnection between the numbers normal to base 2 and the Walsh functions, which are formed by taking products of the Rademacher functions.…”

Section: The Following Results Was Proved Byémile Borel In 1904mentioning

confidence: 99%

Normal Numbers without Measure Theory

Nillsen¹

2000

The American Mathematical Monthly

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“…There seems no published proof for other primes Remark 2. This conjecture is a p-adic version of (1). Thus the conjecture gi point of the vast theory of p-adic modular forms and p-adic Heeke o [7], Dwork [5], Serre [11]). …”

Section: S Akiyamamentioning

confidence: 99%

Analytic Number Theory and Related Topics

Nagasaka¹

1993

Analytic Number Theory and Related Topics

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“…Secondly, many interesting multiplicative functions are pseudo-random (i.e., the spectral measure is continuous). To illustrate this notion, we quote the seminal paper of Mendès France [250] where the following result is proved: Proposition 6.2. -If x ∈ Z q \ N, the Walsh character w x is pseudorandom, but it is not pseudo-random in the sense of Bass [56] (that is, its correlation function does not converge to 0 at infinity).…”

Section: Additive and Multiplicative Functions Sum-of-digits Functionsmentioning

confidence: 99%