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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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Cited by 276 (35 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
A quantitative ergodic theory proof of Szemerédi’s theorem
, 2004
"... A famous theorem of Szemerédi asserts that given any density 0 < δ ≤ 1 and any integer k ≥ 3, any set of integers with density δ will contain infinitely many proper arithmetic progressions of length k. For general k there are essentially four known proofs of this fact; Szemerédi’s original combin ..."
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Cited by 57 (16 self)
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A famous theorem of Szemerédi asserts that given any density 0 < δ ≤ 1 and any integer k ≥ 3, any set of integers with density δ will contain infinitely many proper arithmetic progressions of length k. For general k there are essentially four known proofs of this fact; Szemerédi’s original combinatorial proof using the Szemerédi regularity lemma and van der Waerden’s theorem, Furstenberg’s proof using ergodic theory, Gowers’ proof using Fourier analysis and the inverse theory of additive combinatorics, and Gowers’ more recent proof using a hypergraph regularity lemma. Of these four, the ergodic theory proof is arguably the shortest, but also the least elementary, requiring in particular the use of transfinite induction (and thus the axiom of choice), decomposing a general ergodic system as the weakly mixing extension of a transfinite tower of compact extensions. Here we present a quantitative, selfcontained version of this ergodic theory proof, and which is “elementary ” in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds.
The ergodic and combinatorial approaches to Szemerédi’s theorem
, 2006
"... Abstract. A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graphtheoretical) approac ..."
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Abstract. A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graphtheoretical) approach of Szemerédi, the ergodic theory approach of Furstenberg, the Fourieranalytic approach of Gowers, and the hypergraph approach of NagleRödlSchachtSkokan and Gowers. In this lecture series we introduce the first, second and fourth approaches, though we will not delve into the full details of any of them. One of the themes of these lectures is the strong similarity of ideas between these approaches, despite the fact that they initially seem rather different. 1.
Maximal multilinear operators
 Trans. Amer. Math. Soc
"... Abstract. We establish multilinear L p bounds for a class of maximal multilinear averages of functions on one variable, reproving and generalizing the bilinear maximal function bounds of Lacey [13]. As an application we obtain almost everywhere convergence results for these averages, and in some cas ..."
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Cited by 12 (4 self)
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Abstract. We establish multilinear L p bounds for a class of maximal multilinear averages of functions on one variable, reproving and generalizing the bilinear maximal function bounds of Lacey [13]. As an application we obtain almost everywhere convergence results for these averages, and in some cases we also obtain almost everywhere convergence for their ergodic counterparts on a dynamical system. 1.
Obstructions to uniformity, and arithmetic patterns in the primes
, 2005
"... In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify prec ..."
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Cited by 8 (5 self)
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In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving “randomly”, and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes.
Pointwise characteristic factors for the multiple term return times theorem
, 2003
"... return times theorem ..."
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Arithmetic progressions and the primes
 Collect. Math. (2006
"... We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1. ..."
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Cited by 5 (2 self)
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We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1.
CONVERGENCE OF MULTIPLE ERGODIC AVERAGES ALONG CUBES FOR SEVERAL COMMUTING TRANSFORMATIONS
, 811
"... Abstract. In this paper, we give the convergence result of multiple ergodic averages along cubes for several commuting transformations, and the correspondant combinatorial results. The main tools we use are the seminorms and “magic ” extension introduced by Host recently. 1. ..."
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Abstract. In this paper, we give the convergence result of multiple ergodic averages along cubes for several commuting transformations, and the correspondant combinatorial results. The main tools we use are the seminorms and “magic ” extension introduced by Host recently. 1.