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UNIVERSAL CHARACTERISTIC FACTORS AND FURSTENBERG AVERAGES
, 2004
"... Let X = (X 0, B, µ, T) be an ergodic probability measure preserving system. For a natural number k we consider the averages N ∑ k ∏ 1 fj(T ..."
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Cited by 49 (2 self)
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Let X = (X 0, B, µ, T) be an ergodic probability measure preserving system. For a natural number k we consider the averages N ∑ k ∏ 1 fj(T
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 28 (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.
Multiple ergodic averages for three polynomials and applications
, 2006
"... Abstract. We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form {p, 2p,..., kp}. We then derive several combinatorial implications, including an answer to a question of Brown, ..."
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Cited by 9 (2 self)
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Abstract. We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form {p, 2p,..., kp}. We then derive several combinatorial implications, including an answer to a question of Brown, Graham, and Landman, and a generalization of the Polynomial Szemerédi Theorem of Bergelson and Leibman for families of three polynomials with not necessarily zero constant term. We also simplify and generalize a recent result of Bergelson, Host, and Kra, showing that for all ε> 0 and every subset of the integers Λ the set n ∈ N: d ∗ ( Λ ∩ (Λ + p1(n)) ∩ (Λ + p2(n)) ∩ (Λ + p3(n)) )> (d ∗ (Λ)) 4 − ε} has bounded gaps for “most ” choices of integer polynomials p1, p2, p3. Contents
MULTIPLE RECURRENCE AND CONVERGENCE FOR SEQUENCES RELATED TO THE PRIME NUMBERS
"... Abstract. For any measure preserving system (X, X, µ, T) and A ∈ X with µ(A)> 0, we show that there exist infinitely many primes p such that µ ` A ∩ T −(p−1) A ∩ T −2(p−1) A ´> 0 (the same holds with p − 1 replaced by p + 1). Furthermore, we show the existence of the limit in L 2 (µ) of the as ..."
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Cited by 7 (2 self)
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Abstract. For any measure preserving system (X, X, µ, T) and A ∈ X with µ(A)> 0, we show that there exist infinitely many primes p such that µ ` A ∩ T −(p−1) A ∩ T −2(p−1) A ´> 0 (the same holds with p − 1 replaced by p + 1). Furthermore, we show the existence of the limit in L 2 (µ) of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form p − 1 (or p + 1) for some prime p. 1.
An odd FurstenbergSzemerédi theorem and quasiaffine systems
 J. Analyse Math
"... Abstract. We prove a version of Furstenberg’s ergodic theorem with restrictions on return times. More specifically, for a measure preserving system (X,B, µ, T), an integer 0 ≤ j < k, and E ⊂ X with µ(E)> 0, we show that there exists n ≡ j (mod k) with µ(E∩T−nE∩T−2nE∩T−3nE)> 0, so long as T ..."
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Cited by 4 (1 self)
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Abstract. We prove a version of Furstenberg’s ergodic theorem with restrictions on return times. More specifically, for a measure preserving system (X,B, µ, T), an integer 0 ≤ j < k, and E ⊂ X with µ(E)> 0, we show that there exists n ≡ j (mod k) with µ(E∩T−nE∩T−2nE∩T−3nE)> 0, so long as T k is ergodic. This result requires a deeper understanding of the limit of some non conventional ergodic averages, and the introduction of a new class of systems, the ‘QuasiAffine Systems’. 1.
Extensions of probabilitypreserving systems by measurablyvarying homogeneous spaces and applications
, 2009
"... We study a generalized notion of a homogeneous skewproduct extension of a probabilitypreserving base system in which the homogeneous space fibres can vary over the ergodic decomposition of the base. The construction of such extensions rests on a simple notion of ‘direct integral ’ for a ‘measurabl ..."
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Cited by 1 (1 self)
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We study a generalized notion of a homogeneous skewproduct extension of a probabilitypreserving base system in which the homogeneous space fibres can vary over the ergodic decomposition of the base. The construction of such extensions rests on a simple notion of ‘direct integral ’ for a ‘measurable family’ of homogeneous spaces, which has a number of precedents in older literature. The main contribution of the present paper is the systematic development of a formalism for handling such extensions, including nonergodic versions of the results of Mackey describing ergodic components of such extensions [29], of the FurstenbergZimmer Structure Theory [45, 44, 18] and of results of Mentzen [32] describing the structure of automorphisms of relatively ergodic such extensions. We then