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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.
Generalising the HardyLittlewood method for primes
 In: Proceedings of the international congress of mathematicians
, 2007
"... Abstract. The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the ..."
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Abstract. The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the HardyLittlewood method has been generalised to obtain, for example, an asymptotic for the number of 4term arithmetic progressions of primes less than N.
PRIME POINTS IN THE INTERSECTION OF TWO SUBLATTICES
, 2006
"... Abstract. Let Ψ: Z d → Z t be an affinelinear map, and ˙ Ψ be its linear part. A famous and difficult open conjecture of Dickson predicts the distribution of prime points in the affine sublattice Ψ(Z d) of Z t. Assuming the Gowers Inverse conjecture GI(s) and the Möbius and nilsequences conjecture ..."
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Abstract. Let Ψ: Z d → Z t be an affinelinear map, and ˙ Ψ be its linear part. A famous and difficult open conjecture of Dickson predicts the distribution of prime points in the affine sublattice Ψ(Z d) of Z t. Assuming the Gowers Inverse conjecture GI(s) and the Möbius and nilsequences conjecture MN(s) for some finite s � 1, GreenTao [GT3] verified Dickson’s conjecture for Ψ of complexity s. Let Φ: Z l → Z t be an affinelinear map, and ˙ Φ be its linear part. Suppose that Ψ(Z d) ∩ Φ(Z l) is nonempty and ˙ Ψ(Z d) ∩ ˙ Φ(Z l) is of finite index in ˙ Ψ(Z d). In this paper, as an application of GreenTao’s theorem on Dickson’s conjecture, we study the distribution of prime points in the affine sublattice Ψ(Z d) ∩ Φ(Z l). 1.
SUBLATTICES OF FINITE INDEX
, 2007
"... Abstract. Assuming the Gowers Inverse conjecture and the Möbius conjecture for the finite parameter s, GreenTao verified Dickson’s conjecture for lattices which are ranges of linear maps of complexity at most s. In this paper, we reformulate GreenTao’s theorem on Dickson’s conjecture, and prove th ..."
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Abstract. Assuming the Gowers Inverse conjecture and the Möbius conjecture for the finite parameter s, GreenTao verified Dickson’s conjecture for lattices which are ranges of linear maps of complexity at most s. In this paper, we reformulate GreenTao’s theorem on Dickson’s conjecture, and prove that, if L is the range of a linear map of complexity s, and L1 is a sublattice of L of finite index, then L1 is the range of a linear map of complexity s. 1.