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The primes contain arbitrarily long arithmetic progressions
 Ann. of Math
"... Abstract. We prove that there are arbitrarily long arithmetic progressions of primes. ..."
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
Linear correlations amongst numbers represented by positive definite binary quadratic forms
 Acta Arith
"... Abstract. Let f1,..., ft be positive definite binary quadratic forms, and letRfi(n) = {(x, y) : fi(x, y) = n}  denote the corresponding representation functions. Employing methods developed by Green and Tao, we deduce asymptotics for linear correlations of these representation functions. More pre ..."
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Abstract. Let f1,..., ft be positive definite binary quadratic forms, and letRfi(n) = {(x, y) : fi(x, y) = n}  denote the corresponding representation functions. Employing methods developed by Green and Tao, we deduce asymptotics for linear correlations of these representation functions. More precisely, we study the expression En∈K∩[−N,N]d t∏ i=1 Rfi(ψi(n)), where the ψi form a system of affine linear forms no two of which are affinely related, and where K is a convex body. The minor arc analysis builds on the observation that polynomial subsequences of equidistributed nilsequences are still equidistributed, an observation that could be useful in treating the minor arcs of other arithmetic questions. As a very quick application we give asymptotics to the number of simultaneous zeros of certain systems of quadratic equations in 8 or more variables. Contents
ON MANIN’S CONJECTURE FOR A FAMILY OF CHÂTELET SURFACES
"... Abstract. — The Manin conjecture is established for Châtelet surfaces over Q arising as minimal proper smooth models of the surface Y 2 + Z 2 = f(X) in A3 Q, where f ∈ Z[X] is a totally reducible polynomial of degree 3 without repeated roots. These surfaces do not satisfy weak approximation. Content ..."
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Abstract. — The Manin conjecture is established for Châtelet surfaces over Q arising as minimal proper smooth models of the surface Y 2 + Z 2 = f(X) in A3 Q, where f ∈ Z[X] is a totally reducible polynomial of degree 3 without repeated roots. These surfaces do not satisfy weak approximation. Contents 1. Introduction.................................................
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
 Collect. Math. (2006
"... 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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Cited by 5 (2 self)
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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.
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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Cited by 3 (0 self)
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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.
SUMS OF ARITHMETIC FUNCTIONS OVER VALUES OF BINARY FORMS
, 2006
"... Abstract. Given a suitable arithmetic function h: N → R�0, and a binary form F ∈ Z[x1, x2], we investigate the average order of h as it ranges over the values taken by F. A general upper bound is obtained for this quantity, in which the dependence upon the coefficients of F is made completely explic ..."
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Abstract. Given a suitable arithmetic function h: N → R�0, and a binary form F ∈ Z[x1, x2], we investigate the average order of h as it ranges over the values taken by F. A general upper bound is obtained for this quantity, in which the dependence upon the coefficients of F is made completely explicit. 1.
PAIRS OF DIAGONAL QUADRATIC FORMS AND LINEAR CORRELATIONS AMONG SUMS OF TWO SQUARES
"... Abstract. For suitable pairs of diagonal quadratic forms in 8 variables we use the circle method to investigate the density of simultaneous integer solutions and relate this to the problem of estimating linear correlations among sums of two squares. 1. ..."
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Abstract. For suitable pairs of diagonal quadratic forms in 8 variables we use the circle method to investigate the density of simultaneous integer solutions and relate this to the problem of estimating linear correlations among sums of two squares. 1.