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

Abstract : 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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Contributor : El Houcein El Abdalaoui <>
Submitted on : Wednesday, August 7, 2019 - 9:21:34 AM
Last modification on : Thursday, August 8, 2019 - 1:16:33 AM

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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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