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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.
Norm convergence of multiple ergodic averages for commuting transformations
, 2007
"... Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established fo ..."
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Cited by 81 (4 self)
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Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established for l = 2 by Conze and Lesigne [2] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiT −1 j by Frantzikinakis and Kra [3] (with the l = 3 case of this result established earlier in [29]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l = 2 case of our arguments are a finitary analogue of those in [2].
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.
Lowdegree tests at large distances
, 2006
"... We define tests of boolean functions which distinguish between linear (or quadratic)polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal tradeoffs between soundness and the number of queries. In particular, we show ..."
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Cited by 48 (2 self)
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We define tests of boolean functions which distinguish between linear (or quadratic)polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal tradeoffs between soundness and the number of queries. In particular, we show that functions with small Gowers uniformity norms behave "randomly" with respect to hypergraph linearity tests. A central step in our analysis of quadraticity tests is the proof of an inverse theorem for the third Gowers uniformity norm of boolean functions. The last result has also a coding theory application. It is possible to estimate efficiently the distance from the secondorder ReedMuller code on inputs lying far beyond its listdecoding radius.
On exchangeable random variables and the statistics of large graphs and hypergraphs
, 2008
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The inverse conjecture for the Gowers norm over finite fields via the correspondence principle
, 2010
"... 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 ‖Ud (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 ..."
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Cited by 39 (8 self)
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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 ‖Ud (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, Tao and Ziegler. 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 found so far in this setting can be avoided by a slight reformulation of the conjecture.
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 38 (3 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].
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 ..."
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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.