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31
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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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].
Arithmetic progressions and the primes  El Escorial lectures
 Collectanea Mathematica (2006), Vol. Extra., 3788 (Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial
"... Abstract. We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1. ..."
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Abstract. We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1.
GREENTAO THEOREM IN FUNCTION FIELDS
, 2009
"... We adapt the proof of the GreenTao theorem on arithmetic progressions in primes to the setting of polynomials over a finite field, to show that for every k, the irreducible polynomials in Fq[t] contain configurations of the form {f + Pg: deg(P) < k}, g ̸ = 0. ..."
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We adapt the proof of the GreenTao theorem on arithmetic progressions in primes to the setting of polynomials over a finite field, to show that for every k, the irreducible polynomials in Fq[t] contain configurations of the form {f + Pg: deg(P) < k}, g ̸ = 0.
A RELATIVE SZEMERÉDI THEOREM
"... Abstract. The celebrated GreenTao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemerédi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long ..."
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Abstract. The celebrated GreenTao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemerédi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions. In this paper, we give a simple proof of a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative Szemerédi theorem for kterm arithmetic progressions in pseudorandom subsets of ZN of density N −ck. The key component in our proof is an extension of the regularity method to sparse pseudorandom hypergraphs, which we believe to be interesting in its own right. From this we derive a relative extension of the hypergraph removal lemma. This is a strengthening of an earlier theorem used by Tao in his proof that the Gaussian primes contain arbitrarily shaped constellations and, by standard arguments, allows us to deduce the relative Szemerédi theorem. 1.
ARITHMETIC PROGRESSIONS OF PRIMES IN SHORT INTERVALS
, 705
"... Abstract. Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: If N is sufficiently large and M is not too small compared with N, then the primes in the interval [N, N + M] contains many arithm ..."
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Abstract. Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: If N is sufficiently large and M is not too small compared with N, then the primes in the interval [N, N + M] contains many arithmetic progressions of length k. 1.
PRIME NUMBERS IN LOGARITHMIC INTERVALS
"... Abstract. Let X be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type (p, p + h], where p ≤ X is a prime number and h = o(X). Then we will apply this to prove that for every λ>1/2 there exists a positive proportion of primes p ≤ ..."
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Abstract. Let X be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type (p, p + h], where p ≤ X is a prime number and h = o(X). Then we will apply this to prove that for every λ>1/2 there exists a positive proportion of primes p ≤ X such that the interval (p, p+λ log X] contains at least a prime number. As a consequence we improve Cheer and Goldston’s result on the size of real numbers λ> 1 with the property that there is a positive proportion of integers m ≤ X such that the interval (m, m + λ log X] contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers m ≤ X such that the interval (m, m + λ log X] contains at least a prime number. The last applications of these techniques are two theorems (the first one unconditional and the second one in which we assume the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture) on the positive proportion of primes p ≤ X such that the interval (p, p + λ log X] containsnoprimes. 1.
Additive Combinatorics with a view towards Computer Science and Cryptography  An Exposition
, 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.
A Multiple Sum Involving the Möbius Function
, 2003
"... 1. Introduction. The aim of the present article is to discuss the asymptotics of the quantity Mk(z) = ∑ µ(d1) · · · µ(d2k), k ≥ 1, (1.1) ..."
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1. Introduction. The aim of the present article is to discuss the asymptotics of the quantity Mk(z) = ∑ µ(d1) · · · µ(d2k), k ≥ 1, (1.1)