Results 1 
6 of
6
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 ..."
Abstract

Cited by 55 (9 self)
 Add to MetaCart
(Show Context)
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.
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 27 (1 self)
 Add to MetaCart
(Show Context)
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.
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 non ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
(Show Context)
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.
The polynomial FreimanRuzsa conjecture. http://www.maths.bris.ac.uk/∼mabjg/papers/PFR.pdf. [GT07] [GT08] [KL08] [LMS08] [LN97] [Lov08] [Sam07] [ST06] [Ste03] [Tre09] [TZ08] [Vio08] [VW07
, 2007
"... Abstract. Let G be an abelian group. The Polynomial FreimanRuzsa conjecture (PFR) concerns the structure of sets A ⊆ G for which A + A  6 KA. These notes provide proofs for the statements made in §10 of [8], and as such constitute a reasonably detailed discussion of the PFR in the case G = Fn2. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Let G be an abelian group. The Polynomial FreimanRuzsa conjecture (PFR) concerns the structure of sets A ⊆ G for which A + A  6 KA. These notes provide proofs for the statements made in §10 of [8], and as such constitute a reasonably detailed discussion of the PFR in the case G = Fn2. Although the purpose of these notes is to furnish proofs for the statements in §10 of [8], they are reasonably selfcontained. For further context see the article [8] itself. A great deal of the material in this section was communicated to me in person by Imre Ruzsa, and is reproduced here with his kind permission. 1. tools In this section we assemble a number of tools which are nowadays regarded as part of the standard armoury of an additive combinatorialist. The forthcoming book [18] will serve as a compendium for these and much more besides. Let us briefly recall some notation concerning sumsets. Suppose that G is an abelian group and that A,B ⊆ G. Then we write A+B: = {a+ b  a ∈ A, b ∈ B}. More generally if k, l are any two nonnegative integers then we set kA − lB: = {a1 + · · ·+ ak − b1 − · · · − bl  ai ∈ A, bj ∈ B}. If A  = n and if A+A  6 KA, where K is “small ” relative to n, then we say that A has small doubling. We call the ratio A+ A/A  the doubling constant of A. The first tool is an inequality of Plünnecke [13], a new proof of which was found by Ruzsa [16]. Expositions of the proof may also be found in [12] or [9]. Proposition 1.1 (Plünnecke’s inequalities). Suppose that A and B are subsets of some abelian group G, and that A + B  6 KA. Then for any nonnegative integers k, l we have kB − lB  6 Kk+lA. The second tool is a simple but surprisingly powerful covering lemma of Ruzsa. Lemma 1.2 (Ruzsa). Let S, T be subsets of an abelian group such that S+T  6 K ′S. Then there is a set X ⊆ T, X  6 K ′, such that T ⊆ S − S +X. Proof. Pick a maximal set X ⊆ T such that the sets S+x, x ∈ X, are pairwise disjoint. Since x∈X(S + x) ⊆ S + T, we have SX  6 K ′S, which implies that X  6 K ′. Now suppose that t ∈ T. By maximality we there must be some x ∈ X such that (S + t) ∩ (S + x) 6 = ∅, which means that t ∈ S − S + x.
Abstract. Notes on the sumproduct estimates of BourgainKatzTao and Bourgain
"... The aim of these notes is to give a selfcontained proof of the sumproduct estimates of Bourgain, Katz, Tao and Konyagin. Specifically we will establish: Theorem 1.1. Suppose that p is a prime, and that A ⊆ Fp\{0} is a set with A  � p 1−δ. Then there is an absolute constant c = c(δ)> 0 such t ..."
Abstract
 Add to MetaCart
(Show Context)
The aim of these notes is to give a selfcontained proof of the sumproduct estimates of Bourgain, Katz, Tao and Konyagin. Specifically we will establish: Theorem 1.1. Suppose that p is a prime, and that A ⊆ Fp\{0} is a set with A  � p 1−δ. Then there is an absolute constant c = c(δ)> 0 such that we have the estimate A + A  + A · A  � cA  1+c. 2. The BalogSzemerédiGowers theorem The first three sections are devoted to proving Proposition 4.4. This proposition shows that if A + A  and A · A  are both small then, after passing to a largeish subset of A, we may control more complicated algebraic expressions too. Let A be a subset of an abelian group, written additively. We write M + (A) for the number of additive quadruples in A, that is to say quadruples (a1, a2, a3, a4) ∈ A 4 such that a1 − a2 = a3 − a4. If A is a subset of some abelian group written multiplicatively, then we write M × (A) for the number of solutions to a1/a2 = a3/a4. A very simple application of the CauchySchwarz inequality serves to establish that if A has small doubling, then M + (A) is large. Lemma 2.1. Suppose that A is a subset of an abelian group with A  = N, and that A + A  � KA. Then M + (A) � K −1 N 3. Similarly, suppose that A · A  � KA. Then M × (A) � K −1 N 3.
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 ..."
Abstract
 Add to MetaCart
(Show Context)
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.