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The primes contain arbitrarily long arithmetic progressions
 Ann. of Math
"... Abstract. We prove that there are arbitrarily long arithmetic progressions of primes. ..."
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
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 97 (6 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 dichotomy between structure and randomness, arithmetic progressions, and the primes
"... Abstract. A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness ..."
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Abstract. A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (lowcomplexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the GreenTao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different. 1.
Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems
 Pacific J. Math
"... Abstract. The Furstenberg recurrence theorem (or equivalently, Szemerédi’s theorem) can be formulated in the language of von Neumann algebras as follows: given an integer k ≥ 2, an abelian finite von Neumann algebra (M, τ) with an automorphism α: M → M, and a nonnegative a ∈ M with τ(a)> 0, on ..."
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Abstract. The Furstenberg recurrence theorem (or equivalently, Szemerédi’s theorem) can be formulated in the language of von Neumann algebras as follows: given an integer k ≥ 2, an abelian finite von Neumann algebra (M, τ) with an automorphism α: M → M, and a nonnegative a ∈ M with τ(a)> 0, one has lim infN→ ∞ 1N ∑N n=1 Re τ(aα n(a)... α(k−1)n(a))> 0; a subsequent result of Host and Kra shows that this limit exists. In particular, Re τ(aαn(a)... α(k−1)n(a))> 0 for all n in a set of positive density. From the von Neumann algebra perspective, it is thus natural to ask to what extent these results remain true when the abelian hypothesis is dropped. All three claims hold for k = 2, and we show in this paper that all three claims hold for all k when the von Neumann algebra is asymptotically abelian, and that the last two claims hold for k = 3 when the von Neumann algebra is
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 6 (5 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