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The GreenTao Theorem on arithmetic progressions in the primes: an ergodic point of view
, 2005
"... A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an a ..."
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Cited by 18 (2 self)
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A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory.
Polynomial averages converge to the product of the integrals
 Isr. J. Math
"... Abstract. We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges in L 2 to the product of the integrals. Such averages are characterize ..."
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Cited by 12 (7 self)
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Abstract. We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges in L 2 to the product of the integrals. Such averages are characterized by nilsystems and so we reduce the problem to one of uniform distribution of polynomial sequences on nilmanifolds. 1.
Convergence of multiple ergodic averages for some commuting ransformations
 Erg. Th. & Dyn. Sys
"... Abstract. We prove the L2 convergence for the linear multiple ergodic averages of commuting transformations T1,...,Tl, assuming that each map Ti and each pair TiT −1 j is ergodic for i ̸ = j. The limiting behavior of such averages is controlled by a particular factor, which is an inverse limit of ni ..."
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Cited by 12 (2 self)
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Abstract. We prove the L2 convergence for the linear multiple ergodic averages of commuting transformations T1,...,Tl, assuming that each map Ti and each pair TiT −1 j is ergodic for i ̸ = j. The limiting behavior of such averages is controlled by a particular factor, which is an inverse limit of nilsystems. As a corollary we show that the limiting behavior of linear multiple ergodic averages is the same for commuting transformations.
Convergence of multiple . . .
, 2006
"... These notes are based on a course for a general audience given at the Centro de Modeliamento Matemático of the University of Chile, in December 2004. We study the mean convergence of multiple ergodic averages, that is, averages of a product of functions taken at different times. We also describe th ..."
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These notes are based on a course for a general audience given at the Centro de Modeliamento Matemático of the University of Chile, in December 2004. We study the mean convergence of multiple ergodic averages, that is, averages of a product of functions taken at different times. We also describe the relations between this area of ergodic theory and some classical and some recent results in additive number theory.