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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 suc ..."
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Cited by 30 (4 self)
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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
The Gaussian primes contain arbitrarily shaped constellations
 J. dAnalyse Mathematique
"... Abstract. We show that the Gaussian primes P[i] ⊆ Z[i] contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers v0,..., vk−1, we show that there are infinitely many sets {a+rv0,..., a+rvk−1}, with a ∈ Z[i] and r ∈ Z\{0}, all of ..."
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Cited by 18 (10 self)
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Abstract. We show that the Gaussian primes P[i] ⊆ Z[i] contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers v0,..., vk−1, we show that there are infinitely many sets {a+rv0,..., a+rvk−1}, with a ∈ Z[i] and r ∈ Z\{0}, all of whose elements are Gaussian primes. The proof is modeled on that in [9] and requires three ingredients. The first is a hypergraph removal lemma of Gowers and RödlSkokan, or more precisely a slight strengthening of this lemma which can be found in [22]; this hypergraph removal lemma can be thought of as a generalization of the SzemerédiFurstenbergKatznelson theorem concerning multidimensional arithmetic progressions. The second ingredient is the transference argument from [9], which allows one to extend this hypergraph removal lemma to a relative version, weighted by a pseudorandom measure. The third ingredient is a GoldstonYıldırım type analysis for the Gaussian integers, similar to that in [9], which yields a pseudorandom measure which is concentrated on Gaussian “almost primes”. 1.
The GreenTao Theorem on arithmetic progressions in the primes: an ergodic point of view
, 2005
"... A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an a ..."
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Cited by 18 (2 self)
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A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory.
Obstructions to uniformity, and arithmetic patterns in the primes, preprint
"... Abstract. 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 iden ..."
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Cited by 5 (3 self)
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Abstract. 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. 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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Cited by 2 (0 self)
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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.