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55
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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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.
An improved construction of progressionfree sets
, 2009
"... The problem of constructing dense subsets S of {1, 2,..., n} that contain no threeterm arithmetic progression was introduced by Erdős and Turán in 1936. They have presented a construction with S  = Ω(nlog3 2) elements. Their construction was improved by Salem and Spencer, and further improved by ..."
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Cited by 21 (0 self)
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The problem of constructing dense subsets S of {1, 2,..., n} that contain no threeterm arithmetic progression was introduced by Erdős and Turán in 1936. They have presented a construction with S  = Ω(nlog3 2) elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is
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.
Monochromatic equilateral right triangles on the integer grid
"... For any coloring of the N × N grid using less than log log n colors, one can always find a monochromatic equilateral right triangle, a triangle with vertex coordinates (x, y), (x + d, y), and (x, y + d). ..."
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Cited by 11 (0 self)
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For any coloring of the N × N grid using less than log log n colors, one can always find a monochromatic equilateral right triangle, a triangle with vertex coordinates (x, y), (x + d, y), and (x, y + d).
New bounds for Szemerédi’s theorem, I: Progressions of length 4 in finite field geometries
 Proc. Lond. Math. Soc
"... Abstract. Let k � 3 be an integer, and let G be a finite abelian group with G  = N, where (N, (k − 1)!) = 1. We write rk(G) for the largest cardinality A  of a set A ⊆ G which does not contain k distinct elements in arithmetic progression. The famous theorem of Szemerédi essentially asserts th ..."
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Cited by 10 (5 self)
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Abstract. Let k � 3 be an integer, and let G be a finite abelian group with G  = N, where (N, (k − 1)!) = 1. We write rk(G) for the largest cardinality A  of a set A ⊆ G which does not contain k distinct elements in arithmetic progression. The famous theorem of Szemerédi essentially asserts that rk(Z/NZ) = ok(N). It is known, in fact, that the estimate rk(G) = ok(N) holds for all G. There have been many papers concerning the issue of finding quantitative bounds for rk(G). A result of Bourgain states that r3(G) ≪ N(log log N / logN) 1/2 for all G. In this paper we obtain a similar bound for r4(G) in the particular case G = F n, where F is a fixed finite field with char(F) ̸ = 2, 3 (for example, F = F5). We prove that r4(G) ≪F N(log N) −c for some absolute constant c> 0. In future papers we will treat general abelian groups G, eventually obtaining a comparable result for arbitrary G. 1.
The quantitative behaviour of polynomial orbits on nilmanifolds
, 2007
"... A theorem of Leibman [19] asserts that a polynomial orbit (g(n)Γ)n∈Z on a nilmanifold G/Γ is always equidistributed in a union of closed subnilmanifolds of G/Γ. In this paper we give a quantitative version of Leibman’s result, describing the uniform distribution properties of a finite polynomial o ..."
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Cited by 9 (0 self)
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A theorem of Leibman [19] asserts that a polynomial orbit (g(n)Γ)n∈Z on a nilmanifold G/Γ is always equidistributed in a union of closed subnilmanifolds of G/Γ. In this paper we give a quantitative version of Leibman’s result, describing the uniform distribution properties of a finite polynomial orbit (g(n)Γ) n∈[N] in a nilmanifold. More specifically we show that there is a factorization g = εg ′ γ, where ε(n) is “smooth”, (γ(n)Γ)n∈Z is periodic and “rational”, and (g ′ (n)Γ)n∈P is uniformly distributed (up to a specified error δ) inside some subnilmanifold G ′ /Γ ′ of G/Γ for all sufficiently dense arithmetic progressions P ⊆ [N]. Our bounds are uniform in N and are polynomial in the error tolerance δ. In a subsequent paper [13] we shall use this theorem to establish the Möbius and Nilsequences conjecture from our earlier paper [12].
Zerosum problems in finite abelian groups and affine caps
 QUARTERLY JOURNAL OF MATHEMATICS
, 2007
"... For a finite abelian group G, let s(G) denote the smallest integer l such that every sequence S over G of length S  ≥l has a zerosum subsequence of length exp(G). We derive new upper and lower bounds for s(G), and all our bounds are sharp for special types of groups. The results are not restricte ..."
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Cited by 9 (2 self)
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For a finite abelian group G, let s(G) denote the smallest integer l such that every sequence S over G of length S  ≥l has a zerosum subsequence of length exp(G). We derive new upper and lower bounds for s(G), and all our bounds are sharp for special types of groups. The results are not restricted to groups G of the form G = Cr n, but they respect the structure of the group. In particular, we show s(C4 n) ≥ 20n − 19 for all odd n, which is sharp if n is a power of 3. Moreover, we investigate the relationship between extremal sequences and maximal caps in finite geometry.
Linear equation, arithmetic progressions and hypergraph property testing
 Proc. of the 16 th Annual ACMSIAM SODA, ACM Press
, 2005
"... For a fixed kuniform hypergraph D (kgraph for short, k ≥ 3), we say that a kgraph H) if it contains no copy (resp. induced copy) of D. Our goal in satisfies property PD (resp. P ∗ D this paper is to classify the kgraphs D for which there are propertytesters for testing PD and P ∗ D whose query ..."
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Cited by 8 (2 self)
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For a fixed kuniform hypergraph D (kgraph for short, k ≥ 3), we say that a kgraph H) if it contains no copy (resp. induced copy) of D. Our goal in satisfies property PD (resp. P ∗ D this paper is to classify the kgraphs D for which there are propertytesters for testing PD and P ∗ D whose query complexity is polynomial in 1/ɛ. For such kgraphs we say that PD (resp. P ∗ D) is easily testable. For P ∗ D, we prove that aside from a single 3graph, P ∗ D is easily testable if and only if D is a single kedge. We further show that for large k, one can use more sophisticated techniques in order to obtain better lower bounds for any large enough kgraph. These results extend and improve previous results about graphs [5] and kgraphs [18]. For PD, we show that for any kpartite kgraph D, PD is easily testable, by giving an efficient onesided errorproperty tester, which improves the one obtained by [18]. We further prove a nearly matching lower bound on the query complexity of such a propertytester. Finally, we give a sufficient condition for inferring that PD is not easily testable. Though our results do not supply a complete characterization of the kgraphs for which PD is easily testable, they are a natural