Results 1  10
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10
An improved construction of progressionfree sets
, 2009
"... The problem of constructing dense subsets S of {1, 2,..., n} that contain no threeterm arithmetic progression was introduced by Erdős and Turán in 1936. They have presented a construction with S  = Ω(nlog3 2) elements. Their construction was improved by Salem and Spencer, and further improved by ..."
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Cited by 23 (0 self)
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The problem of constructing dense subsets S of {1, 2,..., n} that contain no threeterm arithmetic progression was introduced by Erdős and Turán in 1936. They have presented a construction with S  = Ω(nlog3 2) elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is
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 ..."
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Cited by 19 (1 self)
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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.
A simple regularization of hypergraphs
"... Abstract. We give a simple and natural construction of hypergraph regularization. It yields a short proof of a hypergraph regularity lemma. Consequently, as an example of its applications, we have a short selfcontained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well a ..."
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Cited by 5 (3 self)
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Abstract. We give a simple and natural construction of hypergraph regularization. It yields a short proof of a hypergraph regularity lemma. Consequently, as an example of its applications, we have a short selfcontained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well as its multidimensional extension by FurstenbergKatznelson (1978). 1.
Threeterm arithmetic progressions and sumsets
 Proc. Edinb. Math. Soc
"... Abstract. Suppose that G is an abelian group and A ⊂ G is finite and contains no nontrivial three term arithmetic progressions. We show that A + A  ≫ε A(log A) 1 3 −ε. 1. ..."
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Abstract. Suppose that G is an abelian group and A ⊂ G is finite and contains no nontrivial three term arithmetic progressions. We show that A + A  ≫ε A(log A) 1 3 −ε. 1.
Bounds on some van der Waerden numbers
 J. Combin. Theory Ser. A
, 2008
"... For positive integers s and k1, k2,..., ks, the van der Waerden number w(k1, k2,..., ks; s) is the minimum integer n such that for every scoloring of {1, 2,..., n}, with colors 1, 2,..., s, there is a kiterm arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k, m ..."
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For positive integers s and k1, k2,..., ks, the van der Waerden number w(k1, k2,..., ks; s) is the minimum integer n such that for every scoloring of {1, 2,..., n}, with colors 1, 2,..., s, there is a kiterm arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k, m; 2) for fixed m. We include a table of values of w(k, 3; 2) that are very close to this lower bound for m = 3. We also give a lower bound for w(k, k,..., k; s) that slightly improves previouslyknown bounds. Upper bounds for w(k, 4; 2) and w(4, 4,..., 4; s) are also provided. 1.
SETS WHOSE DIFFERENCE SET IS SQUAREFREE
"... Abstract. The purpose of this note is to give an exposition of the bestknown bound on the density of sets whose difference set contains no squares which was first derived by Pintz, Steiger and Szemerédi in [PSS88]. We show how their method can be brought in line with the modern view of the energy i ..."
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Abstract. The purpose of this note is to give an exposition of the bestknown bound on the density of sets whose difference set contains no squares which was first derived by Pintz, Steiger and Szemerédi in [PSS88]. We show how their method can be brought in line with the modern view of the energy increment strategy employed in problems such as Szemerédi’s Theorem on arithmetic progressions, and explore the extent to which the particularities of the method are specific to the set of squares. 1.
Additive Combinatorics with a view towards Computer Science and Cryptography An
, 2011
"... 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 ..."
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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 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 crossfertilize 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.
SOLUTIONFREE SETS FOR SUMS OF BINARY FORMS
"... Abstract. In this paper we obtain quantitative estimates for the asymptotic density of subsets of the integer lattice Z 2 which contain only trivial solutions to an additive equation involving binary forms. In the process we develop an analogue of Vinogradov’s mean value theorem applicable to binary ..."
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Abstract. In this paper we obtain quantitative estimates for the asymptotic density of subsets of the integer lattice Z 2 which contain only trivial solutions to an additive equation involving binary forms. In the process we develop an analogue of Vinogradov’s mean value theorem applicable to binary forms. 1.
A PURELY COMBINATORIAL APPROACH TO SIMULTANEOUS POLYNOMIAL RECURRENCE MODULO 1
"... Abstract. Using purely combinatorial means we obtain results on simultaneous Diophantine approximation modulo 1 for systems of polynomials with real coefficients and no constant term. 1. ..."
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Abstract. Using purely combinatorial means we obtain results on simultaneous Diophantine approximation modulo 1 for systems of polynomials with real coefficients and no constant term. 1.