Results 1  10
of
29
The primes contain arbitrarily long arithmetic progressions
 Ann. of Math
"... Abstract. We prove that there are arbitrarily long arithmetic progressions of primes. ..."
Abstract

Cited by 150 (26 self)
 Add to MetaCart
Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
The primes contain arbitrarily long polynomial progressions
 Acta Math
"... Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε suc ..."
Abstract

Cited by 30 (4 self)
 Add to MetaCart
Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε such that x+P1(m),..., x+Pk(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case Pi = (i − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties. Contents
Linear equations in primes
 Annals of Mathematics
"... Abstract. 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 → ..."
Abstract

Cited by 29 (1 self)
 Add to MetaCart
Abstract. 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
On exchangeable random variables and the statistics of large graphs and hypergraphs
, 2008
"... ..."
A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma, preprint
"... Abstract. We introduce a correspondence principle (analogous to the Furstenberg correspondence principle) that allows one to extract an infinite random graph or hypergraph from a sequence of increasingly large deterministic graphs or hypergraphs. As an application we present a new (infinitary) proof ..."
Abstract

Cited by 20 (5 self)
 Add to MetaCart
Abstract. We introduce a correspondence principle (analogous to the Furstenberg correspondence principle) that allows one to extract an infinite random graph or hypergraph from a sequence of increasingly large deterministic graphs or hypergraphs. As an application we present a new (infinitary) proof of the hypergraph removal lemma of NagleSchachtRödlSkokan and Gowers, which does not require the hypergraph regularity lemma and requires significantly less computation. This in turn gives new proofs of several corollaries of the hypergraph removal lemma, such as Szemerédi’s theorem on arithmetic progressions. 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 ..."
Abstract

Cited by 19 (1 self)
 Add to MetaCart
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.
The Gaussian primes contain arbitrarily shaped constellations
 J. dAnalyse Mathematique
"... Abstract. We show that the Gaussian primes P[i] ⊆ Z[i] contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers v0,..., vk−1, we show that there are infinitely many sets {a+rv0,..., a+rvk−1}, with a ∈ Z[i] and r ∈ Z\{0}, all of ..."
Abstract

Cited by 18 (10 self)
 Add to MetaCart
Abstract. We show that the Gaussian primes P[i] ⊆ Z[i] contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers v0,..., vk−1, we show that there are infinitely many sets {a+rv0,..., a+rvk−1}, with a ∈ Z[i] and r ∈ Z\{0}, all of whose elements are Gaussian primes. The proof is modeled on that in [9] and requires three ingredients. The first is a hypergraph removal lemma of Gowers and RödlSkokan, or more precisely a slight strengthening of this lemma which can be found in [22]; this hypergraph removal lemma can be thought of as a generalization of the SzemerédiFurstenbergKatznelson theorem concerning multidimensional arithmetic progressions. The second ingredient is the transference argument from [9], which allows one to extend this hypergraph removal lemma to a relative version, weighted by a pseudorandom measure. The third ingredient is a GoldstonYıldırım type analysis for the Gaussian integers, similar to that in [9], which yields a pseudorandom measure which is concentrated on Gaussian “almost primes”. 1.
The GreenTao Theorem on arithmetic progressions in the primes: an ergodic point of view
, 2005
"... A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an a ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory.
A new proof of the density HalesJewett theorem
, 2009
"... The Hales–Jewett theorem asserts that for every r and every k there exists n such that every rcolouring of the ndimensional grid {1,..., k} n contains a combinatorial line. This result is a generalization of van der Waerden’s theorem, and it is one of the fundamental results of Ramsey theory. The ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
The Hales–Jewett theorem asserts that for every r and every k there exists n such that every rcolouring of the ndimensional grid {1,..., k} n contains a combinatorial line. This result is a generalization of van der Waerden’s theorem, and it is one of the fundamental results of Ramsey theory. The theorem of van der Waerden has a famous density version, conjectured by Erdős and Turán in 1936, proved by Szemerédi in 1975 and given a different proof by Furstenberg in 1977. The Hales–Jewett theorem has a density version as well, proved by Furstenberg and Katznelson in 1991 by means of a significant extension of the ergodic techniques that had been pioneered by Furstenberg in his proof of Szemerédi’s theorem. In this paper, we give the first elementary proof of the theorem of Furstenberg and Katznelson, and the first to provide a quantitative bound on how large n needs to be. In particular, we show that a subset of [3] n of density δ contains a combinatorial line if n ≥ 2 ⇈ O(1/δ 3). Our proof is surprisingly simple: indeed, it gives what is probably the simplest known proof of Szemerédi’s theorem.
The inverse conjecture for the Gowers norm over finite fields via the correspondence principle
 Analysis & PDE
"... Abstract. The inverse conjecture for the Gowers norms U d (V) for finitedimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ‖f ‖ U d (V) if and only if it correlates with a phase polynomial φ = eF(P) of degree at most d − 1, t ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
Abstract. The inverse conjecture for the Gowers norms U d (V) for finitedimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ‖f ‖ U d (V) if and only if it correlates with a phase polynomial φ = eF(P) of degree at most d − 1, thus P: V → F is a polynomial of degree at most d − 1. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case char(F) � d from an ergodic theory counterpart, which was recently established by Bergelson and the authors in [2]. In low characteristic we obtain a partial result, in which the phase polynomial φ is allowed to be of some larger degree C(d). The full inverse conjecture remains open in low characteristic; the counterexamples in [15], [13] in this setting can be avoided by a slight reformulation of the conjecture. 1.