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74
A sumproduct estimate in finite fields, and applications
"... Abstract. Let A be a subset of a finite field F: = Z/qZ for some prime q. If F  δ < A  < F  1−δ for some δ> 0, then we prove the estimate A + A  + A · A  ≥ c(δ)A  1+ε for some ε = ε(δ)> 0. This is a finite field analogue of a result of [ESz1983]. We then use this estimate to ..."
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Cited by 75 (7 self)
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Abstract. Let A be a subset of a finite field F: = Z/qZ for some prime q. If F  δ < A  < F  1−δ for some δ> 0, then we prove the estimate A + A  + A · A  ≥ c(δ)A  1+ε for some ε = ε(δ)> 0. This is a finite field analogue of a result of [ESz1983]. We then use this estimate to prove a SzemerédiTrotter type theorem in finite fields, and obtain a new estimate for the Erdös distance problem in finite fields, as well as the threedimensional Kakeya problem in finite fields. 1.
On the singularity probability of random Bernoulli matrices
 SOC DEPARTMENT OF MATHEMATICS, UCSD, LA JOLLA, CA 92093 EMAIL ADDRESS: KCOSTELL@UCSD.EDU DEPARTMENT OF MATHEMATICS, RUTGERS, PISCATAWAY, NJ 08854 EMAIL ADDRESS: VANVU@MATH.RUTGERS.EDU
, 2005
"... Let n be a large integer and Mn be a random n by n matrix whose entries are i.i.d. Bernoulli random variables (each entry is±1 with probability 1/2). We show that the probability that Mn is singular is at most (3/4+o(1)) n, improving an earlier estimate of Kahn, Komlós and Szemerédi [11], as well a ..."
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Cited by 63 (18 self)
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Let n be a large integer and Mn be a random n by n matrix whose entries are i.i.d. Bernoulli random variables (each entry is±1 with probability 1/2). We show that the probability that Mn is singular is at most (3/4+o(1)) n, improving an earlier estimate of Kahn, Komlós and Szemerédi [11], as well as earlier work by the authors [17]. The key new ingredient is the applications of Freiman type inverse theorems and other tools from additive combinatorics.
Freiman’s theorem in an arbitrary abelian group
, 2006
"... A famous result of Freiman describes the structure of finite sets A ⊆ Z with small doubling property. If A + A  � KA  then A is contained within a multidimensional arithmetic progression of dimension d(K) and size f(K)A. Here we prove an analogous statement valid for subsets of an arbitrary ..."
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Cited by 59 (11 self)
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A famous result of Freiman describes the structure of finite sets A ⊆ Z with small doubling property. If A + A  � KA  then A is contained within a multidimensional arithmetic progression of dimension d(K) and size f(K)A. Here we prove an analogous statement valid for subsets of an arbitrary abelian group.
Product set estimates for noncommutative groups
 COMBINATORICA
, 2006
"... 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 ..."
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Cited by 55 (9 self)
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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.
A PROBABILISTIC TECHNIQUE FOR FINDING ALMOSTPERIODS IN ADDITIVE COMBINATORICS
"... We introduce a new probabilistic technique for finding ‘almostperiods’ of convolutions of subsets of finite groups. This allows us to give probabilistic proofs of two classical results in additive combinatorics: Roth’s theorem on threeterm arithmetic progressions and the existence of long arithmet ..."
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Cited by 32 (3 self)
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We introduce a new probabilistic technique for finding ‘almostperiods’ of convolutions of subsets of finite groups. This allows us to give probabilistic proofs of two classical results in additive combinatorics: Roth’s theorem on threeterm arithmetic progressions and the existence of long arithmetic progressions in sumsets A +B in Zp. The bounds we obtain for these results are not the best ones known—these being established using Fourier analysis—but they are of a somewhat comparable quality, which is unusual for a method that is completely combinatorial. Furthermore, we are able to find long arithmetic progressions in sets A + B even when both A and B have density close to 1 / logp, which is much sparser than has previously been possible.
Sets with small sumset and rectification
 BULL. LONDON MATH. SOC
, 2005
"... We study the extent to which sets A ⊆ Z/NZ, N prime, resemble sets of integers from the additive point of view (“up to Freiman isomorphism”). We give a direct proof of a result of Freiman, namely that if A+A  � KA  and A  < c(K)N then A is Freiman isomorphic to a set of integers. Because ..."
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Cited by 28 (7 self)
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We study the extent to which sets A ⊆ Z/NZ, N prime, resemble sets of integers from the additive point of view (“up to Freiman isomorphism”). We give a direct proof of a result of Freiman, namely that if A+A  � KA  and A  < c(K)N then A is Freiman isomorphic to a set of integers. Because we avoid appealing to Freiman’s structure theorem, we get a reasonable bound: we can take c(K) � (32K) −12K2. As a byproduct of our argument we obtain a sharpening of the second author’s result on, and if A+A  � KA, sets with small sumset in torsion groups. For example if A ⊆ Fn 2 then A is contained in a coset of a subspace of size no more than K222K2 −2 A.
DECOMPOSITIONS, APPROXIMATE STRUCTURE, TRANSFERENCE, AND THE HAHNBANACH THEOREM
, 2008
"... We discuss three major classes of theorems in additive and extremal combinatorics: decomposition theorems, approximate structure theorems, and transference principles. We also show how the finitedimensional HahnBanach theorem can be used to give short and transparent proofs of many results of the ..."
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Cited by 17 (1 self)
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We discuss three major classes of theorems in additive and extremal combinatorics: decomposition theorems, approximate structure theorems, and transference principles. We also show how the finitedimensional HahnBanach theorem can be used to give short and transparent proofs of many results of these kinds. Amongst the applications of this method is a much shorter proof of one of the major steps in the proof of Green and Tao that the primes contain arbitrarily long arithmetic progressions. In order to explain the role of this step, we include a brief description of the rest of their argument. A similar proof has been discovered independently by Reingold, Trevisan, Tulsiani and Vadhan [RTTV].
A quantitative version of the idempotent theorem in harmonic analysis
, 2007
"... Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complexvalued measures under convolution. A measure µ ∈ M(G) is said to be idempotent if µ ∗ µ = µ, or alternatively if ̂µ takes only the values 0 and 1. The CohenHelsonRudin idempotent theorem ..."
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Cited by 16 (7 self)
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Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complexvalued measures under convolution. A measure µ ∈ M(G) is said to be idempotent if µ ∗ µ = µ, or alternatively if ̂µ takes only the values 0 and 1. The CohenHelsonRudin idempotent theorem states that a measure µ is idempotent if and only if the set {γ ∈ ̂ G: ̂µ(γ) = 1} belongs to the coset ring of ̂ G, that is to say we may write L∑
Additive structures in sumsets
"... Abstract. Suppose that A and A ′ are subsets of Z/NZ. We write A + A ′ for the set {a + a ′ : a ∈ A and a ′ ∈ A ′ } and call it the sumset of A and A ′. In this paper we address the following question. Suppose that A1,...,Am are subsets of Z/NZ. Does A1 +... + Am contain a long arithmetic progressi ..."
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Cited by 14 (4 self)
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Abstract. Suppose that A and A ′ are subsets of Z/NZ. We write A + A ′ for the set {a + a ′ : a ∈ A and a ′ ∈ A ′ } and call it the sumset of A and A ′. In this paper we address the following question. Suppose that A1,...,Am are subsets of Z/NZ. Does A1 +... + Am contain a long arithmetic progression? The situation for m = 2 is rather different from that for m ≥ 3. In the former case we provide a new proof of a result due to Green. He proved that A1+ A2 contains an arithmetic progression of length roughly exp(c √ α1α2 log N) where α1 and α2 are the respective densities of A1 and A2. In the latter case we improve the existing estimates. For example we show that if A ⊂ Z/NZ has density α ≫ √ log log N/log N then A + A + A contains an arithmetic progression of length Ncα. This compares with the previous best of Ncα2+ε. Two main ingredients have gone into the paper. The first is the observation that one can apply the iterative method to these problems using some machinery of Bourgain. The second is that we can localize a result due to Chang regarding the large spectrum of L2functions. This localization seems to be of interest in its own right and has already found one application elsewhere. 1.