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Pré-Publication, Document De Travail Année : 2019

Simple proof of Bourgain bilinear ergodic theorem and its extension to polynomials and polynomials in primes

Une preuve simple du théorème ergodique bilinéaire de Bourgain et son extension aux polynômes et polynômes premiers.

Résumé

We first present a modern simple proof of the classical ergodic Birkhoff's theorem and Bourgain's homogeneous bilinear ergodic theorem. This proof used the simple fact that the shift map on integers has a simple Lebesgue spectrum. As a consequence , we establish that the homogeneous bilinear ergodic averages along polynomials and polynomials in primes converge almost everywhere, that is, for any invertible measure preserving transformation T , acting on a probability space (X, B, µ), for any f ∈ L r (X, µ) , g ∈ L r ′ (X, µ) such that 1/r + 1/r′ = 1, for any nonconstant polynomials P(n), Q(n), n ∈ Z, taking integer values, and for almost all x ∈ X, we have, 1/N ΣN n=1 f(T^P(n) x) and 1/(πN) Σp≤N p prime f(T^P (p)x)g(T^Q(p)x), converge. Here πN is the number of prime in [1, N].
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Dates et versions

hal-02264446 , version 1 (07-08-2019)

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  • HAL Id : hal-02264446 , version 1

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El Houcein El Abdalaoui. Simple proof of Bourgain bilinear ergodic theorem and its extension to polynomials and polynomials in primes. 2019. ⟨hal-02264446⟩
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