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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 151 (26 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
Roth’s Theorem in the primes
 Annals of Math
"... Abstract. We show that any set containing a positive proportion of the primes contains a 3term arithmetic progression. An important ingredient is a proof that the primes enjoy the socalled HardyLittlewood majorant property. We derive this by giving a new proof of a rather more general result of B ..."
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Cited by 21 (4 self)
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Abstract. We show that any set containing a positive proportion of the primes contains a 3term arithmetic progression. An important ingredient is a proof that the primes enjoy the socalled HardyLittlewood majorant property. We derive this by giving a new proof of a rather more general result of Bourgain which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes. 1.
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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Cited by 19 (1 self)
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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.
Obstructions to uniformity, and arithmetic patterns in the primes, preprint
"... Abstract. 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 iden ..."
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Abstract. 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. 1.
Arithmetic structures in random sets
, 2008
"... We extend two wellknown results in additive number theory, Sárközy’s theorem on square differences in dense sets and a theorem of Green on long arithmetic progressions in sumsets, to subsets of random sets of asymptotic density 0. Our proofs rely on a restrictiontype Fourier analytic argument of G ..."
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Cited by 4 (0 self)
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We extend two wellknown results in additive number theory, Sárközy’s theorem on square differences in dense sets and a theorem of Green on long arithmetic progressions in sumsets, to subsets of random sets of asymptotic density 0. Our proofs rely on a restrictiontype Fourier analytic argument of Green and GreenTao. 1
Arithmetic progressions and the primes  El Escorial lectures
 Collectanea Mathematica (2006), Vol. Extra., 3788 (Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial
"... Abstract. 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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Abstract. 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.
Long arithmetic progressions of primes
 Mathematics Proceedings
"... Abstract. This is an article for a general mathematical audience on the author’s work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. 1. introduction and history This is a description of recent work of the author and Terence Tao [11] on primes ..."
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Abstract. This is an article for a general mathematical audience on the author’s work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. 1. introduction and history This is a description of recent work of the author and Terence Tao [11] on primes in arithmetic progression. It is based on seminars given for a general mathematical
ARBITRARILY LONG ARITHMETIC PROGRESSIONS OF PRIMES ON THE NOSE
, 2005
"... Abstract. Let iL denote the inverse to the logarithmic integral function, Li and define ˆπ(x) to be the number of primes p ≤ x with p = [iL(n)] for some n. We say that these primes hit the value suggested by the prime number theorem “on the nose”. It is known [6] that ˆπ(x) ∼ x log 2 x (so notice in ..."
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Abstract. Let iL denote the inverse to the logarithmic integral function, Li and define ˆπ(x) to be the number of primes p ≤ x with p = [iL(n)] for some n. We say that these primes hit the value suggested by the prime number theorem “on the nose”. It is known [6] that ˆπ(x) ∼ x log 2 x (so notice in particular that the sum of the reciprocals of primes on the nose converges). Combining the theorem of Green and Tao on long arithmetic progressions of primes and estimates of exponential sums, we show that there are arbitrarily long arithmetic progressions of primes on the nose. This is the first sequence of primes to contain long APs and have a converging sum of reciprocals. 1.
THREE TOPICS IN ADDITIVE PRIME NUMBER THEORY
, 710
"... Abstract. We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: wor ..."
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Abstract. We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes. Introduction. These notes are to accompany two lectures I am scheduled to give at the Current Developments in Mathematics conference at Harvard in November 2007. The title of those lectures is ‘A good new millennium for primes’, but I have chosen a rather drier title for these notes for two reasons. Firstly, the title of the lectures was unashamedly stolen (albeit with permission) from Andrew Granville’s entertaining
CHEN’S PRIMES AND TERNARY GOLDBACH PROBLEM
, 812
"... and Abstract. We prove that there exists a k0> 0 such that every sufficiently large odd integer n with 3  n can be represented as p1 + p2 + p3, where p1, p2 are Chen’s primes and p3 is a prime with p3 + 2 has at most k0 prime factors. 1. ..."
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and Abstract. We prove that there exists a k0> 0 such that every sufficiently large odd integer n with 3  n can be represented as p1 + p2 + p3, where p1, p2 are Chen’s primes and p3 is a prime with p3 + 2 has at most k0 prime factors. 1.