Finite and infinite arithmetic progressions in sumsets

Szemerédi, Endre; Vu, Van

doi:10.4007/annals.2006.163.1

Cited by 36 publications

(37 citation statements)

References 15 publications

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“…Section 9 discusses a conjecture of Erdös (posed in 1962) which is related to the above-mentioned conjecture of Folkman. This conjecture was proved in an earlier paper [28] using a special case of the main result in Section 7, but here we give a shorter proof using the general condition worked out in Section 6. Several other applications of the main result of this part will appear in future papers [30,31].…”

Section: Overviewmentioning

confidence: 95%

“…In a previous paper [28], we proved Conjecture 9.2. However, we discuss this problem here for pedagogical reasons.…”

Section: Erdös's Conjecture On Complete Sequencesmentioning

confidence: 99%

“…The verification of the structural lemma occupies most of Section 7. Section 8 contains the rest of the proof, whose core consists of an observation about proper GAPs (subsection 8.3) and a variant of the so-called tiling technique, introduced in an earlier paper [28]. Section 9 discusses a conjecture of Erdös (posed in 1962) which is related to the above-mentioned conjecture of Folkman.…”

Section: Overviewmentioning

confidence: 99%

“…In order to find a triplet (A , l , n ) as desired, we are going to apply the so-called tree argument. This argument was introduced in [28] and, in spirit, works as follows. Assume that we want to add several sets A 1 , .…”

Section: Filling With Different Setsmentioning

confidence: 99%

“…The paper contains several new technical ingredients, some of which (such as the study of proper GAPs in Sections 3 and 7 and the rank reduction argument used in Sections 3 and 4) would be of independent interest. Our writing benefits from two earlier papers [27,28], which established several partial results and launched the foundation of our study. Many ideas from these two papers will be used here, frequently in more general and more comprehensible forms.…”

Section: Overviewmentioning

confidence: 99%

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Long arithmetic progressions in sumsets: Thresholds and bounds

Szemerédi

2005

J. Amer. Math. Soc.

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For a set A A of integers, the sumset l A = A + ⋯ + A lA =A+\dots +A consists of those numbers which can be represented as a sum of l l elements of A A : \[ l A = { a 1 + ⋯ + a l | a i ∈ A i } . lA =\{a_1+\dots + a_l| a_i \in A_i \}. \] Closely related and equally interesting notion is that of l ∗ A l^{\ast }A , which is the collection of numbers which can be represented as a sum of l l different elements of A A : \[ l ∗ A = { a 1 + ⋯ + a l | a i ∈ A i , a i ≠ a j } . l^{\ast }A =\{a_1+\dots + a_l| a_i \in A_i, a_i \neq a_j \}. \] The goal of this paper is to investigate the structure of l A lA and l ∗ A l^{\ast }A , where A A is a subset of { 1 , 2 , … , n } \{1,2, \dots , n\} . As application, we solve two conjectures by Erdös and Folkman, posed in 1960s.

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Section: Overviewmentioning

confidence: 95%

“…In a previous paper [28], we proved Conjecture 9.2. However, we discuss this problem here for pedagogical reasons.…”

Section: Erdös's Conjecture On Complete Sequencesmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

Section: Filling With Different Setsmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

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Long arithmetic progressions in sumsets: Thresholds and bounds

Szemerédi

2005

J. Amer. Math. Soc.

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The Thirties

Narkiewicz

2012

Springer Monographs in Mathematics

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Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

Bibak

2013

Springer Proceedings in Mathematics &Amp; Statistics

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Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define -perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of combinatorial properties of algebraic objects, for instance, Abelian groups, rings, or fields. This emerging field has seen tremendous advances over the last few years, and has recently become a focus of attention among both mathematicians and computer scientists. This fascinating area has been enriched by its formidable links to combinatorics, number theory, harmonic analysis, ergodic theory, and some other branches; all deeply cross-fertilize each other, holding great promise for all of them! In this exposition, we attempt to provide an overview of some breakthroughs in this field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography.

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Finite and infinite arithmetic progressions in sumsets

Cited by 36 publications

References 15 publications

Long arithmetic progressions in sumsets: Thresholds and bounds

Long arithmetic progressions in sumsets: Thresholds and bounds

The Thirties

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

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