Additive Combinatorics

We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et al. [Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes. Proceedings of the 43rd annual ACM symposium on Theory of computing, STOC 11, (ACM, NY 2011), 519-528] in which they were used to answer questions regarding point configurations. In this work, we derive near-optimal rank bounds for these matrices and use them to obtain asymptotically tight bounds in many of the geometric applications. As a consequence of our improved analysis, we also obtain a new, linear algebraic, proof of Kelly's theorem, which is the complex analog of the Sylvester-Gallai theorem.

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“…LEMMA 1.15 (Freiman's Lemma. See [27]). Let A be a finite subset of R d , and suppose that |A + A| K |A|.…”

Section: A Variation Onmentioning

confidence: 99%

Improved Rank Bounds for Design Matrices and a New Proof of Kelly’s Theorem

Dvir¹,

Saraf

Wigderson

2014

Forum math. Sigma

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“…Freiman's celebrated theorem gives a classification of the sets with small doubling K : they are dense subsets of generalized arithmetic progressions of rank at most K . See [28] for the precise result and its history. Theorems 2.2 and 2.4 roughly state that, if K < 3, then A is efficiently covered by an arithmetic progression.…”

Section: Generalization Of Freiman's 3k − 3 Theorem To Popular Sumsmentioning

confidence: 99%

“…The statement and proof of the Cauchy-Davenport-Chowla theorem can be found in [28,Ch. 5], and this robust version is contained in [19].…”

Section: Introductionmentioning

confidence: 99%

An -Function-Free Proof of Vinogradov’s Three Primes Theorem

Shao

2014

Forum math. Sigma

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We give a new proof of Vinogradov's three primes theorem, which asserts that all sufficiently large odd positive integers can be written as the sum of three primes. Existing proofs rely on the theory of L-functions, either explicitly or implicitly. Our proof is sieve theoretical and uses a transference principle, the idea of which was first developed by Green [Ann. of Math. (2008), . To make our argument work, we also develop an additive combinatorial result concerning popular sums, which may be of independent interest.

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“…Kneser generalized this result to an arbitrary abelian group by proving the following [15][14] [13][16] [21].…”

Section: Introductionmentioning

confidence: 99%

“…The case t = 1 was previously treated in [10]. The proof makes use of the notion of additive energy (see [21][10]), which we will introduce in the proof. …”

mentioning

confidence: 99%

On extending Pollard’s theorem for t-representable sums

Grynkiewicz

2010

Isr. J. Math.

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Abstract. Let t ≥ 1, let A and B be finite, nonempty subsets of an abelian group G, and let A + i B denote all the elements c with at least i representations of the form c = a + b, with a ∈ A and b ∈ B. For |A|, |B| ≥ t, we show that eitheror else there exist A ′ ⊆ A and B ′ ⊆ B withwhere H is the (nontrivial) stabilizer of A + t B and ρ = |A ′ + H| − |A ′ | + |B ′ + H| − |B ′ |.In the case t = 2, we improve (1) to |A + 1 B| + |A + 2 B| ≥ 2|A| + 2|B| − 4.

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Additive Combinatorics

Cited by 958 publications

References 0 publications

Improved Rank Bounds for Design Matrices and a New Proof of Kelly’s Theorem

Improved Rank Bounds for Design Matrices and a New Proof of Kelly’s Theorem

An -Function-Free Proof of Vinogradov’s Three Primes Theorem

On extending Pollard’s theorem for t-representable sums

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