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41
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 34 (1 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].
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 ..."
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Cited by 30 (4 self)
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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
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.
A Removal Lemma for Systems of Linear Equations over Finite Fields
, 2008
"... We prove a removal lemma for systems of linear equations over finite fields: let X1,..., Xm be subsets of the finite field Fq and let A be a (k×m) matrix with coefficients in Fq and rank k; if the linear system Ax = b has o(q m−k) solutions with xi ∈ Xi, then we can destroy all these solutions by de ..."
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Cited by 15 (0 self)
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We prove a removal lemma for systems of linear equations over finite fields: let X1,..., Xm be subsets of the finite field Fq and let A be a (k×m) matrix with coefficients in Fq and rank k; if the linear system Ax = b has o(q m−k) solutions with xi ∈ Xi, then we can destroy all these solutions by deleting o(q) elements from each Xi. This extends a result of Green [Geometric and Functional Analysis 15(2) (2005), 340–376] for a single linear equation in abelian groups to systems of linear equations. In particular, we also obtain an analogous result for systems of equations over integers, a result conjectured by Green. Our proof uses the colored version of the hypergraph Removal Lemma.
Weak hypergraph regularity and linear hypergraphs
, 2009
"... We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any kuniform hypergraph H of positive uniform density contains all linear kuniform hypergraphs of a given size. More precisely, we show that for all integers ℓ ≥ k ≥ 2 and every d> ..."
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Cited by 14 (6 self)
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We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any kuniform hypergraph H of positive uniform density contains all linear kuniform hypergraphs of a given size. More precisely, we show that for all integers ℓ ≥ k ≥ 2 and every d> 0 there exists ϱ> 0 for which the following holds: if H is a sufficiently large kuniform hypergraph with the property that the density of H induced on every vertex subset of size ϱn is at least d, then H contains every linear kuniform hypergraph F with ℓ vertices. The main ingredient in the proof of this result is a counting lemma for linear hypergraphs, which establishes that the straightforward extension of graph εregularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems.
The ergodic and combinatorial approaches to Szemerédi’s theorem
, 2006
"... Abstract. A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graphtheoretical) approac ..."
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Cited by 12 (2 self)
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Abstract. A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graphtheoretical) approach of Szemerédi, the ergodic theory approach of Furstenberg, the Fourieranalytic approach of Gowers, and the hypergraph approach of NagleRödlSchachtSkokan and Gowers. In this lecture series we introduce the first, second and fourth approaches, though we will not delve into the full details of any of them. One of the themes of these lectures is the strong similarity of ideas between these approaches, despite the fact that they initially seem rather different. 1.
Property testing in hypergraphs and the removal lemma (Extended Abstract)
, 2006
"... Property testers are efficient, randomized algorithms which recognize if an input graph (or other combinatorial structure) satisfies a given property or if it is “far” from exhibiting it. Generalizing several earlier results, Alon and Shapira showed that hereditary graph properties are testable (wit ..."
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Cited by 10 (0 self)
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Property testers are efficient, randomized algorithms which recognize if an input graph (or other combinatorial structure) satisfies a given property or if it is “far” from exhibiting it. Generalizing several earlier results, Alon and Shapira showed that hereditary graph properties are testable (with onesided error). In this paper we prove the analogous result for hypergraphs. This result is an immediate consequence of a (hyper)graph theoretic statement, which is an extension of the socalled removal lemma. The proof of this generalization relies on the regularity method for hypergraphs.
A proof of Green’s conjecture regarding the removal properties of sets of linear equations
"... A system of ℓ linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1,..., n} which contains o(n p−ℓ) solutions of Mx = b can be turned into a set S ′ containing no solution of Mx = b, by the removal of o(n) elements. Green [GAFA 2005] proved that a single homo ..."
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Cited by 8 (1 self)
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A system of ℓ linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1,..., n} which contains o(n p−ℓ) solutions of Mx = b can be turned into a set S ′ containing no solution of Mx = b, by the removal of o(n) elements. Green [GAFA 2005] proved that a single homogenous linear equation always has the removal property, and conjectured that every set of homogenous linear equations has the removal property. In this paper we confirm Green’s conjecture by showing that every set of linear equations (even nonhomogenous) has the removal property. We also discuss some applications of our result in theoretical computer science, and in particular, use it to resolve a conjecture of Bhattacharyya, Chen, Sudan and Xie [7] related to algorithms for testing properties of boolean functions.