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Obstructions to uniformity, and arithmetic patterns in the primes
, 2005
"... In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify prec ..."
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In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving “randomly”, and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes.
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.
Journal de Théorie des Nombres
, 2005
"... Restriction theory of the Selberg sieve, with applications ..."
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)
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.
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.
NORM FORMS FOR ARBITRARY NUMBER FIELDS AS PRODUCTS OF LINEAR POLYNOMIALS
"... Abstract. Given a number field K/Q and a polynomial P ∈ Q[t], all of whose roots are in Q, let X be the variety defined by the equation NK(x) = P (t). Combining additive combinatorics with descent we show that the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak appr ..."
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Abstract. Given a number field K/Q and a polynomial P ∈ Q[t], all of whose roots are in Q, let X be the variety defined by the equation NK(x) = P (t). Combining additive combinatorics with descent we show that the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of X. Contents