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252
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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Cited by 276 (35 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
Szemerédi's Regularity Lemma and Its Applications in Graph Theory
, 1996
"... Szemerédi's Regularity Lemma is an important tool in discrete mathematics. It says that, in some sense, all graphs can be approximated by randomlooking graphs. Therefore the lemma helps in proving theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. Recent ..."
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Cited by 262 (3 self)
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Szemerédi's Regularity Lemma is an important tool in discrete mathematics. It says that, in some sense, all graphs can be approximated by randomlooking graphs. Therefore the lemma helps in proving theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. Recently quite a few new results were obtained by using the Regularity Lemma, and also some new variants and generalizations appeared. In this survey we describe some typical applications and some generalizations.
Nonconventional ergodic averages and nilmanifolds
"... Abstract. We study the L2convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions appearing in Furstenberg’s proof of Szemerédi’s Theorem. The second average is taken along cube ..."
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Cited by 154 (16 self)
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Abstract. We study the L2convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions appearing in Furstenberg’s proof of Szemerédi’s Theorem. The second average is taken along cubes whose sizes tend to +∞. For each average, we show that it is sufficient to prove the convergence for special systems, the characteristic factors. We build these factors in a general way, independent of the type of the average. To each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold. From the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density. 1.
An ergodic Szemer'edi theorem for commuting transformations
 J. Analyse Math
, 1979
"... The classical Poincar6 recurrence theorem asserts that under the action of a measure preserving transformation T of a finite measure space (X, ~, p.), every set A of positive measure recurs in the sense that for some n> 0,/z (T'A n A)> 0. In [1] this was extended to multiple recurrence: ..."
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Cited by 114 (2 self)
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The classical Poincar6 recurrence theorem asserts that under the action of a measure preserving transformation T of a finite measure space (X, ~, p.), every set A of positive measure recurs in the sense that for some n> 0,/z (T'A n A)> 0. In [1] this was extended to multiple recurrence: the transformations T, T2,..., T k have a common power satisfying /x (A n ThA n... n Tk"A)> 0 for a set A of positive measure. We also showed that this result implies Szemer6di's theorem stating that any set of integers of positive upper density contains arbitrarily long arithmetic progressions. In [2] a topological analogue of this is proved: if T is a homeomorphism of a compact metric space X, for any e>0 and k = 1,2,3,.., there is a point x E X and a common power of T, T 2, 9 9 9 T k such that d(x, Tnx) < e, d(x, T2"x) < e,. 9 d(x, Tk~x) < e. This (weaker) result, in turn, implies van der Waerden's theorem on arithmetic progressions for partitions of the integers. Now in this case a virtually identical argument shows that the topological result is true for any k commuting transformations. This would lead one to expect that the measure theoretic result is also true for arbitrary commuting transformations. (It is easy to give a counterexample with noncommuting transformations.) We prove this in what follows. Theorem A. Let (X,~,/z) be a measure space with /z(X)<oo, let T~, T2, 9 9 9 Tk be commuting measure preserving transformations of X and let A E B with tz (A)> O. Then lim inf 1 N ~N 1 Iz ( T~'A A Tj~A N...A T~A)>O. A corollary is the multidimensional extension of Szemer6di's theorem: Theorem B. Let S C Z " be a subset with positive upper density and let F C Z " be any finite configuration. Then there exists an integer d and a vector n E Z " such that n+dFCS.
The counting lemma for regular kuniform hypergraphs
, 2004
"... Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓpartite graph with V (G) = V1 ∪ · · · ∪ Vℓ and Vi  = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are εregular of ..."
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Cited by 108 (14 self)
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Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓpartite graph with V (G) = V1 ∪ · · · ∪ Vℓ and Vi  = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are εregular of density d for ℓ 1 ≤ i < j ≤ ℓ, then G contains (1 ± fℓ(ε))d
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 93 (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
Norm convergence of multiple ergodic averages for commuting transformations
, 2007
"... Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established fo ..."
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Cited by 81 (4 self)
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Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established for l = 2 by Conze and Lesigne [2] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiT −1 j by Frantzikinakis and Kra [3] (with the l = 3 case of this result established earlier in [29]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l = 2 case of our arguments are a finitary analogue of those in [2].
On the superrigidity of malleable actions with spectral gap
 J. Amer. Math. Soc
"... Abstract. We prove that if a countable group Γ contains a nonamenable subgroup with centralizer infinite and “weakly normal ” in Γ (e.g. if Γ is nonamenable and has infinite center or is a product of infinite groups) then any measure preserving Γaction on a probability space which satisfies certa ..."
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Cited by 79 (7 self)
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Abstract. We prove that if a countable group Γ contains a nonamenable subgroup with centralizer infinite and “weakly normal ” in Γ (e.g. if Γ is nonamenable and has infinite center or is a product of infinite groups) then any measure preserving Γaction on a probability space which satisfies certain malleability, spectral gap and weak mixing conditions is cocycle superrigid. We also show that if Γ � X is an arbitrary free ergodic action of such a group Γ and Λ � Y = T Λ is a Bernoulli action of an arbitrary infinite conjugacy class group, then any isomorphism of the associated II1 factors L ∞ X ⋊Γ ≃ L ∞ Y ⋊Λ comes from a conjugacy of the actions. 1.
A variant of the hypergraph removal lemma
, 2006
"... Abstract. Recent work of Gowers [10] and Nagle, Rödl, Schacht, and Skokan [15], [19], [20] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [26] and FurstenbergKatznelson [7] concerning onedimensional and multidimensional arithmetic progressions respecti ..."
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Cited by 77 (7 self)
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Abstract. Recent work of Gowers [10] and Nagle, Rödl, Schacht, and Skokan [15], [19], [20] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [26] and FurstenbergKatznelson [7] concerning onedimensional and multidimensional arithmetic progressions respectively. In this paper we shall give a selfcontained proof of this hypergraph removal lemma. In fact we prove a slight strengthening of the result, which we will use in a subsequent paper [29] to establish (among other things) infinitely many constellations of a prescribed shape in the Gaussian primes. 1.
A POLYNOMIAL BOUND IN FREIMAN’S THEOREM
 DUKE MATHEMATICAL JOURNAL VOL. 113, NO. 3
, 2002
"... In this paper the following improvement on Freiman’s theorem on set addition is obtained (see Theorems 1 and 2 in Section 1). Let A ⊂ Z be a finite set such that A + A  < αA. Then A is contained in a proper ddimensional progression P, where d ≤ [α − 1] and log(P/A) < Cα 2 (log α) 3. E ..."
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Cited by 75 (3 self)
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In this paper the following improvement on Freiman’s theorem on set addition is obtained (see Theorems 1 and 2 in Section 1). Let A ⊂ Z be a finite set such that A + A  < αA. Then A is contained in a proper ddimensional progression P, where d ≤ [α − 1] and log(P/A) < Cα 2 (log α) 3. Earlier bounds involved exponential dependence in α in the second estimate. Our argument combines I. Ruzsa’s method, which we improve in several places, as well as Y. Bilu’s proof of Freiman’s theorem.