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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 268 (31 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
Growth and generation in SL2(Z/pZ)
 ANN. OF MATH
, 2005
"... We show that every subset of SL2(Z/pZ) grows rapidly under multiplication. It follows readily that, for every set of generators A of SL2(Z/pZ), every element of SL2(Z/pZ) can be expressed as a product of at most O((log p) c) elements of A ∪ A −1, where c and the implied constant are absolute. ..."
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Cited by 80 (6 self)
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We show that every subset of SL2(Z/pZ) grows rapidly under multiplication. It follows readily that, for every set of generators A of SL2(Z/pZ), every element of SL2(Z/pZ) can be expressed as a product of at most O((log p) c) elements of A ∪ A −1, where c and the implied constant are absolute.
Linear equations in primes
 ANNALS OF MATHEMATICS
, 2006
"... Consider a system Ψ of nonconstant affinelinear forms ψ1,..., ψt: Z d → Z, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [−N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N → ∞, for th ..."
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Cited by 79 (3 self)
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Consider a system Ψ of nonconstant affinelinear forms ψ1,..., ψt: Z d → Z, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [−N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N → ∞, for the number of integer points n ∈ Z d ∩ K for which the integers ψ1(n),..., ψt(n) are simultaneously prime. This implies many other wellknown conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture. It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime. In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affinelinear forms ψ1,..., ψt are affinely related; this excludes the important “binary ” cases such as the twin prime or Goldbach conjectures, but does allow one to count “nondegenerate ” configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the inverse Gowersnorm conjecture (GI(s)) and the Möbius and nilsequences conjecture (MN(s)), where s ∈ {1, 2,...} is
A sumproduct estimate in finite fields, and applications
"... Abstract. Let A be a subset of a finite field F: = Z/qZ for some prime q. If F  δ < A  < F  1−δ for some δ> 0, then we prove the estimate A + A  + A · A  ≥ c(δ)A  1+ε for some ε = ε(δ)> 0. This is a finite field analogue of a result of [ESz1983]. We then use this estimate to ..."
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Cited by 75 (7 self)
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Abstract. Let A be a subset of a finite field F: = Z/qZ for some prime q. If F  δ < A  < F  1−δ for some δ> 0, then we prove the estimate A + A  + A · A  ≥ c(δ)A  1+ε for some ε = ε(δ)> 0. This is a finite field analogue of a result of [ESz1983]. We then use this estimate to prove a SzemerédiTrotter type theorem in finite fields, and obtain a new estimate for the Erdös distance problem in finite fields, as well as the threedimensional Kakeya problem in finite fields. 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 74 (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.
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 55 (15 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.
Product set estimates for noncommutative groups
 COMBINATORICA
, 2006
"... We develop the PlünneckeRuzsa and BalogSzemerédiGowers theory of sum set estimates in the noncommutative setting, with discrete, continuous, and metric entropy formulations of these estimates. We also develop a Freimantype inverse theorem for a special class of 2step nilpotent groups, namely ..."
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Cited by 55 (9 self)
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We develop the PlünneckeRuzsa and BalogSzemerédiGowers theory of sum set estimates in the noncommutative setting, with discrete, continuous, and metric entropy formulations of these estimates. We also develop a Freimantype inverse theorem for a special class of 2step nilpotent groups, namely the Heisenberg groups with no 2torsion in their vertical group.
Extracting randomness using few independent sources
 In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
, 2004
"... In this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2. Specifically, for every δ> 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/δ) number of distributions over ..."
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Cited by 50 (6 self)
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In this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2. Specifically, for every δ> 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/δ) number of distributions over {0, 1} n, each having minentropy δn. These extractors output n bits, which are 2 −n close to uniform. This construction uses several results from additive number theory, and in particular a recent one by Bourgain, Katz and Tao [BKT03] and of Konyagin [Kon03]. We also consider the related problem of constructing randomness dispersers. For any constant output length m, our dispersers use a constant number of identical distributions, each with minentropy Ω(log n) and outputs every possible mbit string with positive probability. The main tool we use is a variant of the “steppingup lemma ” used in establishing lower bound
Turán Numbers of Bipartite Graphs and Related RamseyType Questions
, 2003
"... For a graph H and an integer n, theTurán number ex(n, H) isthemaximum possible number of edges in a simple graph on n vertices that contains no copy of H. H is rdegenerate if every one of its subgraphs contains a vertex of degree at most r. Weprove that, for any fixed bipartite graph H in which all ..."
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Cited by 48 (22 self)
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For a graph H and an integer n, theTurán number ex(n, H) isthemaximum possible number of edges in a simple graph on n vertices that contains no copy of H. H is rdegenerate if every one of its subgraphs contains a vertex of degree at most r. Weprove that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, ex(n, H) �O(n 2−1/r). This is tight for all values of r and can also be derived from an earlier result of Füredi. We also show that there is an absolute positive constant c such that, for every fixed bipartite rdegenerate graph H, ex(n, H) �O(n 1−c/r). This is motivated by aconjectureofErdős that asserts that, for every such H, ex(n, H) �O(n 1−1/r). For two graphs G and H, the Ramsey number r(G, H) istheminimum number n such that, in any colouring of the edges of the complete graph on n vertices by red and blue, there is either aredcopy of G or a blue copy of H. Erdős conjectured that there is an absolute constant c such that, for any graph G with m edges, r(G, G) �2 c √ m.Hereweprove this conjecture for bipartite graphs G, andprove that for general graphs G with m edges,