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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.
Product set estimates for noncommutative groups
 Combinatorica
"... Abstract. 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 ..."
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Cited by 17 (3 self)
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Abstract. 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. 1.
Compressions, Convex Geometry and the FreimanBilu Theorem, preprint
"... Abstract. We note a link between combinatorial results of Bollobás and Leader concerning sumsets in the grid, the BrunnMinkowski theorem and a result of Freiman and Bilu concerning the structure of sets A ⊆ Z with small doubling. Our main result is the following. If ε> 0 and if A is a finite nonemp ..."
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Cited by 2 (1 self)
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Abstract. We note a link between combinatorial results of Bollobás and Leader concerning sumsets in the grid, the BrunnMinkowski theorem and a result of Freiman and Bilu concerning the structure of sets A ⊆ Z with small doubling. Our main result is the following. If ε> 0 and if A is a finite nonempty subset of a torsionfree abelian group with A + A  � KA, then A may be covered by e KO(1) progressions of dimension ⌊log 2 K + ε ⌋ and size at most A. 1.
SET ADDITION IN BOXES AND THE FREIMANBILU THEOREM
, 2005
"... Abstract. Suppose that A is a subset of the box d∏ Q = Q(L1,...,Ld): = {0, 1,..., Li − 1} with A  = αL1... Ld, where L1 � L2 �... � Ld � 1 are integers. We prove that if then A + A  � 2 d/48 A. i=1 α> (d/Ld) 1/2d By combining this with Chang’s quantitative version of Freiman’s theorem, we pr ..."
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Abstract. Suppose that A is a subset of the box d∏ Q = Q(L1,...,Ld): = {0, 1,..., Li − 1} with A  = αL1... Ld, where L1 � L2 �... � Ld � 1 are integers. We prove that if then A + A  � 2 d/48 A. i=1 α> (d/Ld) 1/2d By combining this with Chang’s quantitative version of Freiman’s theorem, we prove a structural result about sets with small sumset. If A ⊆ Z has A + A  � KA, then there is a progression P of dimension O(log K) such that A ∩ P  � exp(−K O(1))max(A, P ). This is closely related to a theorem of Freiman and Bilu, but is quantitatively stronger in certain aspects. 1.