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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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Cited by 276 (35 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
The primes contain arbitrarily long polynomial progressions
 Acta Math
"... Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε ..."
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Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε such that x+P1(m),..., x+Pk(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case Pi = (i − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties. Contents
Restriction theory of Selberg’s sieve, with applications, to appear, Journal de Theorie de Nombres de Bordeaux
"... Abstract. The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L 2 –L p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime ktuples. Let a1,..., ak and b1,. ..."
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Abstract. The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L 2 –L p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime ktuples. Let a1,..., ak and b1,...,bk be positive integers. Write h(θ): = ∑ n∈X e(nθ), where X is the set of all n � N such that the numbers a1n + b1,..., akn + bk are all prime. We obtain upper bounds for ‖h ‖ L p (T), p> 2, which are (conditionally on the prime tuple conjecture) of the correct order of magnitude. As a second application we deduce from Chen’s theorem, Roth’s theorem, and a transference principle that there are infinitely many arithmetic progressions p1 < p2 < p3 of primes, such that pi + 2 is either a prime or a product of two primes for each i = 1, 2, 3.
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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Cited by 8 (5 self)
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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.
An explicit result of the sum of seven cubes ∗
, 2007
"... We prove that every integer ≥ exp(524) is a sum of seven non negative cubes. 1 History and statements In his 1770’s ”Meditationes Algebraicae”, E.Waring asserted that every positive integer is a sum of nine nonnegative cubes. A proof was missing, as was fairly common at the time, the very notion of ..."
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We prove that every integer ≥ exp(524) is a sum of seven non negative cubes. 1 History and statements In his 1770’s ”Meditationes Algebraicae”, E.Waring asserted that every positive integer is a sum of nine nonnegative cubes. A proof was missing, as was fairly common at the time, the very notion of proof being not so clear. Notice that henceforth, we shall use cubes to denote cubes of nonnegative integers. Consequently, the integers we want to write as sums of cubes are assumed to be nonnegative. Maillet in [15] proved that twentyone cubes were enough to represent every (nonnegative) integer and later, Wieferich in [30] provided a proof to Waring’s statement (though his proof contained a mistake that was mended in [12]). The Göttingen school was in full bloom and Landau [13] showed that eight cubes suffice to represent every large enough integer. Dickson [7]
SUMS OF POSITIVE DENSITY SUBSETS OF THE PRIMES
"... Abstract. We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A + B in the natural numbers is at least (1 − o(1))α/(eγ log log(1/β)) which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeni ..."
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Abstract. We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A + B in the natural numbers is at least (1 − o(1))α/(eγ log log(1/β)) which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work the problem is reduced to a similar problem for subsets of Z ∗ m using techniques of Green and GreenTao. Concerning this new problem we show that, for any squarefree m and any A, B ⊆ Z ∗ m of densities α and β, the density of A + B in Zm is at least (1 − o(1))α/(eγ log log(1/β)), which is asymptotically best possible when m is a product of small primes. We also discuss an inverse question. 1.
1 A REMARK ON THE GEOMETRY OF SPACES OF FUNCTIONS WITH PRIME FREQUENCIES.
"... Abstract. For any positive integer r, denote by Pr the set of all integers γ ∈ Z having at most r prime divisors. We show CPr (T), the space of all continuous functions on the circle T whose Fourier spectrum lies in Pr, contains a complemented copy of ℓ1. In particular, CPr (T) is not isomorphic to ..."
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Abstract. For any positive integer r, denote by Pr the set of all integers γ ∈ Z having at most r prime divisors. We show CPr (T), the space of all continuous functions on the circle T whose Fourier spectrum lies in Pr, contains a complemented copy of ℓ1. In particular, CPr (T) is not isomorphic to C(T), nor to the disc algebra A(D). A similar result holds in the L1 setting.
Journal de Théorie des Nombres de Bordeaux 18 (2006), 147–182
"... Restriction theory of the Selberg sieve, with applications ..."
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