## Additive Combinatorics with a view towards Computer Science and Cryptography -- An Exposition (2011)

### BibTeX

@MISC{Bibak11additivecombinatorics,

author = {Khodakhast Bibak},

title = {Additive Combinatorics with a view towards Computer Science and Cryptography -- An Exposition },

year = {2011}

}

### OpenURL

### Abstract

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 additive structures in sets equipped with a group structure – we may have other structure that interacts with this group structure. This newly 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! There is a considerable number of incredible problems, results, and novel applications in this thriving area. In this exposition, we attempt to provide an illuminating overview of some conspicuous breakthroughs in this captivating field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography.

### Citations

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Citation Context ...nd the references therein); the latter have themselves several applications in many areas of number theory, combinatorics, computer science, mathematical and theoretical physics, chemistry, and so on =-=[1, 49, 51, 100, 162, 185, 226, 227]-=-. Also, many prominent applications to group theory, analysis, exponential sums, expanders, complexity theory, and gripping results in discrete geometry, dynamical systems, and various other scientifi... |

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Citation Context ... combinatorics, discrete geometry, and theoretical computer science. One astounding application of this lemma is to the arena of property testing, which is now a very dynamic area in computer science =-=[4, 5, 6, 9, 96, 97, 131, 174, 177, 181, 182, 188]-=-. Property testing typically refers to the existence of sub-linear time probabilistic algorithms (called testers), which distinguish between objects G (e.g., a graph) having a given property P (e.g., ... |

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Citation Context ...232], states that every subset A of integers with positive upper density, that is, lim sup N→∞ |A ∩ [1, N]|/N > 0, has arbitrary long arithmetic progressions. A stunning breakthrough of Green and Tao =-=[111]-=- (that answers a long-standing and folkloric conjecture by Erdős on arithmetic progressions, in a special case: the primes) says that primes contain arbitrary long arithmetic progressions. The fusion ... |

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Citation Context ...newly emerging and captivating area of mathematics has recently found a great deal of remarkable applications to computer science and cryptography, for example, to the arenas of randomness extractors =-=[13, 14, 17, 28, 70, 126, 195, 196]-=-, pseudorandomness [20, 152, 234, 237, 238], property testing [96, 188], complexity theory [26, 165], hardness amplification [239, 240], probabilistic checkable proofs (PCPs) [189], information theory... |

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Citation Context ...integers. More precisely, the Erdős-Turán conjecture states that if δ and k are given, then 3there is a number N = N(k, δ) such that any set A ⊆ [1, N] with |A| ≥ δN contains a nontrivial k-AP. Roth =-=[184]-=- employed methods from Fourier analysis (or more specifically, the Hardy-Littlewood circle method) to prove the k = 3 case of the Erdős-Turán conjecture (see also [19, 22, 30, 67, 73, 120, 145, 158, 1... |

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Citation Context ...g theory to estimating Davenport constants), are often extremely sophisticated, but most results have a simple formulation. For example, a celebrated result by Szemerédi, known as Szemerédi’s theorem =-=[7, 8, 18, 83, 102, 103, 115, 117, 160, 167, 175, 176, 179, 180, 212, 215, 216, 232]-=-, states that every subset A of integers with positive upper density, that is, lim sup N→∞ |A ∩ [1, N]|/N > 0, has arbitrary long arithmetic progressions. A stunning breakthrough of Green and Tao [111... |

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Citation Context ...ter with some difficulties in applying the Szemerédi-Trotter incidence theorem in this setting. In fact, the crossing lemma, which is an important ingredient in the proof of Szemerédi-Trotter theorem =-=[210]-=-, relies on Euler’s formula (and so on the topology of the plane), and consequently does not work in finite fields. Note that the proof that Szemerédi and Trotter presented for their theorem was somew... |

106 | A new proof for Szemeredi’s theorem for arithmetic progressions of length four. Geometric and Functional Analysis
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Citation Context ...inish problems on structures in special sets of zero density (such as the primes) to problems on sets of positive density in the integers. Gowers [102] generalized the arguments previously studied in =-=[101, 184]-=-), in a substantial way. In fact, he employed combinatorics, generalized Fourier analysis, and inverse arithmetic combinatorics (including multilinear versions of Freiman’s theorem on sumsets, and the... |

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Citation Context ...ld, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography. 1 Introduction Additive combinatorics =-=[224]-=- is a compelling and fast growing area of research in mathematical sciences, and the goal of this paper is to survey some of the recent developments and notable accomplishments of the field, focusing ... |

93 |
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Citation Context ...g theory to estimating Davenport constants), are often extremely sophisticated, but most results have a simple formulation. For example, a celebrated result by Szemerédi, known as Szemerédi’s theorem =-=[7, 8, 18, 83, 102, 103, 115, 117, 160, 167, 175, 176, 179, 180, 212, 215, 216, 232]-=-, states that every subset A of integers with positive upper density, that is, lim sup N→∞ |A ∩ [1, N]|/N > 0, has arbitrary long arithmetic progressions. A stunning breakthrough of Green and Tao [111... |

89 | Expanders that beat the eigenvalue bound: explicit construction and application
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Citation Context ...dure is to decompose them into a structured part and a pseudorandom part. Constructions of randomness extractors have been used to get constructions of communication networks and good expander graphs =-=[53, 244]-=-, error correcting codes [121, 230], cryptographic protocols [153, 236], data structures [159] and samplers [246]. Randomness 13extractors are used widely in cryptographic applications (see [61, 138,... |

88 | A characterization of the (natural) graph properties testable with onesided error
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(Show Context)
Citation Context ... combinatorics, discrete geometry, and theoretical computer science. One astounding application of this lemma is to the arena of property testing, which is now a very dynamic area in computer science =-=[4, 5, 6, 9, 96, 97, 131, 174, 177, 181, 182, 188]-=-. Property testing typically refers to the existence of sub-linear time probabilistic algorithms (called testers), which distinguish between objects G (e.g., a graph) having a given property P (e.g., ... |

86 |
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Citation Context ...and a complementary arithmetic counting lemma that have several applications, in particular, an astonishing proof of Szemerédi’s theorem. The triangle removal lemma established by Ruzsa and Szemerédi =-=[187]-=- is one of the most notable applications of Szemerédi’s regularity lemma, asserting that each graph of order n with o(n 3 ) triangles can be made triangle-free by removing o(n 2 ) edges. In other word... |

83 | On data structures and asymmetric communication complexity
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Citation Context ...ess extractors have been used to get constructions of communication networks and good expander graphs [53, 244], error correcting codes [121, 230], cryptographic protocols [153, 236], data structures =-=[159]-=- and samplers [246]. Randomness 13extractors are used widely in cryptographic applications (see [61, 138, 245]); for example, in extracting many private bits even when the adversary knows all except ... |

82 | On the density of some sequences of integers
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(Show Context)
Citation Context ... consequences in Ramsey theory, of which van der Waerden’s theorem and its multidimensional version, i.e., the Gallai-Witt theorem (see, e.g., [91, 107, 136] for further information). Erdős and Turán =-=[76]-=- proposed a very strong form of van der Waerden’s theorem – the density version of van der Waerden’s theorem. They conjectured that arbitrarily long APs appear not only in finite partitions but also i... |

81 |
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Citation Context ...as the sum-product theorem for Fp. In fact, this theorem holds if A is not too close to be the whole field. The condition |A| ≥ p δ in this theorem was removed by Bourgain, Glibichuk, and Konyagin in =-=[41]-=-. Also, note that the condition |A| ≤ p 1−δ is necessary (e.g., if we consider a set A consisting of all elements of the field except one, then max{|A + A|, |A · A|} = |A| + 1). The idea for the proof... |

78 |
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Citation Context ...d by k! + 1. There are three fundamental ingredients in the proof of the Green-Tao theorem (in fact, there are many similarities between Green and Tao’s approach and the ergodic-theoretic method, see =-=[132]-=-). The first is Szemerédi’s theorem itself. Since the primes do not have positive upper density, Szemerédi’s theorem cannot be directly applied. The second major ingredient in the proof is a certain t... |

75 | An ergodic Szemerédi theorem for commuting transformations
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Citation Context ...le. This ergodic-theoretic method is one of the most flexible known proofs, and has been very prosperous at reaching considerable generalizations of Szemerédi’s theorem. Furstenberg and Katznelson in =-=[84]-=- obtained the multidimensional Szemerédi theorem; their proof relies on the concept of multiple recurrence, a powerful tool in the interaction between ergodic theory and additive combinatorics. A pure... |

67 | Constructing locally computable extractors and cryptosystems in the bounded-storage model
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Citation Context ...rt. Constructions of randomness extractors have been used to get constructions of communication networks and good expander graphs [53, 244], error correcting codes [121, 230], cryptographic protocols =-=[153, 236]-=-, data structures [159] and samplers [246]. Randomness 13extractors are used widely in cryptographic applications (see [61, 138, 245]); for example, in extracting many private bits even when the adve... |

66 |
Randomness conductors and constant-degree lossless expanders
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(Show Context)
Citation Context ...dure is to decompose them into a structured part and a pseudorandom part. Constructions of randomness extractors have been used to get constructions of communication networks and good expander graphs =-=[53, 244]-=-, error correcting codes [121, 230], cryptographic protocols [153, 236], data structures [159] and samplers [246]. Randomness 13extractors are used widely in cryptographic applications (see [61, 138,... |

66 | The counting lemma for regular k-uniform hypergraphs
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Citation Context |

66 | Regularity lemma for k-uniform hypergraphs
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Citation Context |

56 |
More on the sum-product phenomenon in prime fields and its applications
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Citation Context ... result estimates the cardinality of #Γp(T ) of the set Γp(T ) = {γ ∈ Fp : ord γ ≤ T and ord(γ + γ −1 ) ≤ T }. As another application of the sum-product theorem, Bourgain, Katz and Tao [43] (also see =-=[25]-=-) derived an important Szemerédi-Trotter type theorem in prime finite fields: If Fp is a prime field, and P and L are points and lines in the projective plane over Fp with cardinality |P|, |L| ≤ N < p... |

54 |
A sum-product estimate in finite
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(Show Context)
Citation Context ...ter case we can take A to be a subring which leads to the degenerate case |A| = |A + A| = |A · A|. A stunning result in the case finite 10field Fp, with p prime, was proved by Bourgain, Katz and Tao =-=[43]-=-. Indeed, they proved the following: if A ⊂ Fp, and p δ ≤ |A| ≤ p 1−δ for some δ > 0, then there exists ε = ε(δ) > 0 such that max{|A + A|, |A · A|} ≥ c|A| 1+ε . This result is now known as the sum-pr... |

52 |
On triples in arithmetic progression
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(Show Context)
Citation Context ...ontains a nontrivial k-AP. Roth [184] employed methods from Fourier analysis (or more specifically, the Hardy-Littlewood circle method) to prove the k = 3 case of the Erdős-Turán conjecture (see also =-=[19, 22, 30, 67, 73, 120, 145, 158, 163, 190, 192]-=-). It is worth mentioning that a Roth-type result has been obtained in [143], asserting that every compact set of reals with Lebesgue measure zero supporting a probabilistic measure satisfying appropr... |

51 |
On sets of integers containing no four elements in arithmetic progression
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(Show Context)
Citation Context ...every compact set of reals with Lebesgue measure zero supporting a probabilistic measure satisfying appropriate dimensionality and Fourier decay conditions must contain non-trivial 3APs. Szemerédi in =-=[211]-=- verified the Erdős-Turán conjecture for arithmetic progressions of length four. Finally, Szemerédi in [212] by a tour de force of ingenious and sophisticated combinatorial argument proved the conject... |

49 |
Randomness-optimal oblivious sampling. Random Structures and Algorithms
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(Show Context)
Citation Context ... been used to get constructions of communication networks and good expander graphs [53, 244], error correcting codes [121, 230], cryptographic protocols [153, 236], data structures [159] and samplers =-=[246]-=-. Randomness 13extractors are used widely in cryptographic applications (see [61, 138, 245]); for example, in extracting many private bits even when the adversary knows all except log Ω(1) n of the n... |

48 | Extracting randomness using few independent sources
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(Show Context)
Citation Context ...newly emerging and captivating area of mathematics has recently found a great deal of remarkable applications to computer science and cryptography, for example, to the arenas of randomness extractors =-=[13, 14, 17, 28, 70, 126, 195, 196]-=-, pseudorandomness [20, 152, 234, 237, 238], property testing [96, 188], complexity theory [26, 165], hardness amplification [239, 240], probabilistic checkable proofs (PCPs) [189], information theory... |

48 | Gowers uniformity, influence of variables, and pcps
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(Show Context)
Citation Context ...7, 28, 70, 126, 195, 196], pseudorandomness [20, 152, 234, 237, 238], property testing [96, 188], complexity theory [26, 165], hardness amplification [239, 240], probabilistic checkable proofs (PCPs) =-=[189]-=-, information theory [155, 156, 223, 235], discrete logarithm based range protocols [54], and Diffie-Hellman distributions [23, 24, 52, 82]. Additive combinatorics has also important applications in e... |

47 |
On the number of sums and products
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(Show Context)
Citation Context ... a small but positive ε, where A is a subset of the reals. They also conjectured that max{|A + A|, |A · A|} ≥ c|A| 2−δ , for any positive δ. Much efforts have been made towards the value of ε. Elekes =-=[72]-=- observed that the sum-product problem has interesting connections to problems in incidence geometry. In particular, he applied the so-called Szemerédi-Trotter theorem and showed that ε ≥ 1/4, if A is... |

47 | A variant of the hypergraph removal lemma
- Tao
(Show Context)
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45 |
Rigorous location of phase transitions in hard optimization problems
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Citation Context ...strong techniques for studying the socalled threshold phenomena, which is itself of significant importance in combinatorics, computer science, discrete probability, statistical physics, and economics =-=[2, 21, 42, 80, 81, 137]-=-. There are also very strong connections between ideas of additive combinatorics and the theory of random matrices (see [225, 226, 241] and the references therein); the latter have themselves several ... |

44 |
A density version of the Hales-Jewett theorem
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(Show Context)
Citation Context ...77, 219]). Also, Austin in [7] proved it via both ergodic-theoretic and combinatorial approaches. The multidimensional Szemerédi theorem was significantly generalized by Furstenberg and Katznelson in =-=[85]-=- (via ergodic-theoretic approaches), and Austin in [8] (via both ergodic-theoretic and combinatorial approaches), to the density Hales-Jewett 5theorem. The density Hales-Jewett theorem states that fo... |

43 | Every monotone graph property is testable
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(Show Context)
Citation Context ... combinatorics, discrete geometry, and theoretical computer science. One astounding application of this lemma is to the arena of property testing, which is now a very dynamic area in computer science =-=[4, 5, 6, 9, 96, 97, 131, 174, 177, 181, 182, 188]-=-. Property testing typically refers to the existence of sub-linear time probabilistic algorithms (called testers), which distinguish between objects G (e.g., a graph) having a given property P (e.g., ... |

41 | Extractor codes
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(Show Context)
Citation Context ...tructured part and a pseudorandom part. Constructions of randomness extractors have been used to get constructions of communication networks and good expander graphs [53, 244], error correcting codes =-=[121, 230]-=-, cryptographic protocols [153, 236], data structures [159] and samplers [246]. Randomness 13extractors are used widely in cryptographic applications (see [61, 138, 245]); for example, in extracting ... |

40 | Simulating Independence: New Constructions of Condensers, Ramsey Graphs, Dispersers, and Extractors. Preprint available at http://www.math.ias.edu/ boaz/Papers/BKSSW.html
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(Show Context)
Citation Context ...newly emerging and captivating area of mathematics has recently found a great deal of remarkable applications to computer science and cryptography, for example, to the arenas of randomness extractors =-=[13, 14, 17, 28, 70, 126, 195, 196]-=-, pseudorandomness [20, 152, 234, 237, 238], property testing [96, 188], complexity theory [26, 165], hardness amplification [239, 240], probabilistic checkable proofs (PCPs) [189], information theory... |

39 | Pseudorandom Bits for Polynomials
- Bogdanov, Viola
- 2010
(Show Context)
Citation Context ... recently found a great deal of remarkable applications to computer science and cryptography, for example, to the arenas of randomness extractors [13, 14, 17, 28, 70, 126, 195, 196], pseudorandomness =-=[20, 152, 234, 237, 238]-=-, property testing [96, 188], complexity theory [26, 165], hardness amplification [239, 240], probabilistic checkable proofs (PCPs) [189], information theory [155, 156, 223, 235], discrete logarithm b... |

38 | Extractors for a constant number of polynomially small min-entropy independent sources
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(Show Context)
Citation Context ...tors are used widely in cryptographic applications (see [61, 138, 245]); for example, in extracting many private bits even when the adversary knows all except log Ω(1) n of the n bits [171] (see also =-=[170]-=-). They also have remarkable applications to quantum cryptography, where photons are used by the randomness extractor to generate secure random bits [195]. Ramsey graphs (that is, graphs that have no ... |

38 | Random matrices: universality of local eigenvalue statistics
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(Show Context)
Citation Context ...nd the references therein); the latter have themselves several applications in many areas of number theory, combinatorics, computer science, mathematical and theoretical physics, chemistry, and so on =-=[1, 49, 51, 100, 162, 185, 226, 227]-=-. Also, many prominent applications to group theory, analysis, exponential sums, expanders, complexity theory, and gripping results in discrete geometry, dynamical systems, and various other scientifi... |

37 | On sums and products of integers
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(Show Context)
Citation Context ... set of reals C = {c1, . . . , cn} is called convex if ci+1 −ci > ci −ci−1, for all i). In the reals setting, does there exist an A ⊂ R for which max{|A + A|, |A · A|} is ‘small’? Erdős and Szemerédi =-=[75]-=- gave a negative answer to this question. Actually, they proved the inequality max{|A+A|, |A·A|} ≥ c|A| 1+ε for a small but positive ε, where A is a subset of the reals. They also conjectured that max... |

36 |
Extremal problems on set systems, Random Structure and Algorithms
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(Show Context)
Citation Context ...-free by removing o(n 2 ) edges. Very recently, Fox [78] gave a marvelous proof of the graph removal lemma which avoids applying Szemerédi’s regularity lemma and gives a better bound. Frankl and Rödl =-=[79]-=- showed that a removal lemma for k-uniform hypergraphs could be applied to prove Szemerédi’s theorem on (k + 1)-APs (recall that a k-uniform hypergraph H = (V, E) has a vertex set V and an edge set E,... |

35 | Non-asymptotic theory of random matrices: Extreme singular values
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(Show Context)
Citation Context ...nd the references therein); the latter have themselves several applications in many areas of number theory, combinatorics, computer science, mathematical and theoretical physics, chemistry, and so on =-=[1, 49, 51, 100, 162, 185, 226, 227]-=-. Also, many prominent applications to group theory, analysis, exponential sums, expanders, complexity theory, and gripping results in discrete geometry, dynamical systems, and various other scientifi... |

34 |
Extremal Combinatorics with Applications in Computer Science
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- 2001
(Show Context)
Citation Context ...eks combinatorial lines. This theorem has many intriguing consequences in Ramsey theory, of which van der Waerden’s theorem and its multidimensional version, i.e., the Gallai-Witt theorem (see, e.g., =-=[91, 107, 136]-=- for further information). Erdős and Turán [76] proposed a very strong form of van der Waerden’s theorem – the density version of van der Waerden’s theorem. They conjectured that arbitrarily long APs ... |

34 | Low-degree tests at large distances
- Samorodnitsky
(Show Context)
Citation Context ...applications to computer science and cryptography, for example, to the arenas of randomness extractors [13, 14, 17, 28, 70, 126, 195, 196], pseudorandomness [20, 152, 234, 237, 238], property testing =-=[96, 188]-=-, complexity theory [26, 165], hardness amplification [239, 240], probabilistic checkable proofs (PCPs) [189], information theory [155, 156, 223, 235], discrete logarithm based range protocols [54], a... |

33 | Applications of the regularity lemma for uniform hypergraphs, submitted
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(Show Context)
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33 | A quantitative ergodic theory proof of Szemerédi’s theorem
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(Show Context)
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32 |
On set of integers which contain no three in arithmetic progression
- Behrend
- 1946
(Show Context)
Citation Context ...uppose rk(N) is the cardinality of the largest subset of [1, N] containing no nontrivial k-APs. Giving asymptotic estimates on rk(N) is an important inverse problem in additive combinatorics. Behrend =-=[16]-=- proved that ( N r3(N) = Ω 2 2√2 √ ) . log2 N 1/4 . log N Rankin [169] generalized Behrend’s construction to longer APs. Roth proved that r3(N) = o(N). In fact, he proved the first nontrivial upper bo... |