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## The primes contain arbitrarily long arithmetic progressions

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Venue: | Ann. of Math |

Citations: | 268 - 31 self |

### Citations

250 |
Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions,
- Furstenberg
- 1977
(Show Context)
Citation Context ... theorem can now be proved in several ways. The original proof of Szemerédi [37, 38] was combinatorial. In 1977, Furstenberg made a very important breakthrough by providing an ergodic-theoretic proof =-=[10]-=-. Perhaps surprisingly for a result about primes, our paper has at least as much in common with the ergodic-theoretic approach as it does with the harmonic analysis approach of Gowers. We will use a l... |

226 |
Some problems of ‘partitio numerorum’ III, On the expression of a number as a sum of primes
- Hardy, Littlewood
- 1922
(Show Context)
Citation Context ...simple heuristic based on the prime number theorem would suggest that there are ≫ N 2 / log k N ktuples of primes p1, . . .,pk in arithmetic progression, each pi being at most N. Hardy and Littlewood =-=[24]-=-, in their famous paper of 1923, advanced a very general conjecture which, as a special case, contains the hypothesis that the number of such k-term progressions is asymptotically CkN 2 / log k N for ... |

221 | On certain sets of integers. - Roth - 1953 |

150 | A new proof of Szemeredi’s theorem for arithmetic progressions of length four. Geometric and Functional Analysis, - Gowers - 1998 |

125 |
Polynomial extensions of van der Waerden’s and Szemeredi’s theorems,
- Bergelson, Leibman
- 1996
(Show Context)
Citation Context ...or more generally inside any subset of a pseudorandom set (such as the “almost primes”) of positive relative density. Thus, for instance, one is led to conjecture a Bergelson-Leibman type result (cf. =-=[4]-=-) for primes. That is, one could hope to show that if Fi : N → N are polynomials with F(0) = 0, then there are infinitely many configurations (a + F1(d), . . ., a + Fk(d)) in which all k elements are ... |

113 | An ergodic Szemeredi theorem for commuting transformations,
- Furstenberg, Katznelson
- 1978
(Show Context)
Citation Context ... modification to our current argument, in large part because of the need to truncate the step parameter d to be at most a small power of N. In a similar spirit, the work of Furstenberg and Katznelson =-=[11]-=- on multidimensional analogues of Szemerédi’s theorem, combined with this transference principle, now suggests that one should be able to show that the Gaussian primes in Z[i] contain infinitely many ... |

113 | The Theory of the Riemann Zeta Function - Titchmarsh |

108 | On some sequences of integers
- Erdős, Turán
- 1936
(Show Context)
Citation Context ...bound is N0(δ, 3) � 2 Cδ−2 log(1/δ) , a result of Bourgain [6]. The hypothetical bound (2.1) is closely related to the following very open conjecture of Erdős and Turán: Conjecture 2.2 (Erdős-Turán). =-=[9]-=- Suppose that A = {a1 < a2 < . . . } is an infinite sequence of integers such that ∑ 1/ai = ∞. Then A contains arbitrarily long arithmetic progressions. This would imply Theorem 1.1. We do not make pr... |

91 | Universal characteristic factors and Furstenberg averages,
- Ziegler
- 2007
(Show Context)
Citation Context ... equations in sets of integers) and certain parts of ergodic theory. Particularly exciting is the suspicion that the notion of a k-step nilsystem, explored in many ergodic-theoretical works (see e.g. =-=[27, 28, 29, 44]-=-), might be analogous to a kind of “higher order Fourier analysis” which could be used to deal with systems of linear equations that cannot be handled by conventional Fourier analysis (a simple exampl... |

82 | A density version of the Hales–Jewett theorem.
- Furstenberg, Katznelson
- 1991
(Show Context)
Citation Context ... show that the Gaussian primes in Z[i] contain infinitely many constellations of any prescribed shape, and similarly for other number fields. Furthermore, the later work of Furstenberg and Katznelson =-=[12]-=- on density Hales-Jewett theorems suggests that one could also show that for any finite field F, the monic irreducible polynomials in F[t] contain affine subspaces over F of arbitrarily high dimension... |

69 |
On sets of integers containing no four elements in arithmetic progression.
- Szemeredi
- 1969
(Show Context)
Citation Context ... a statement about sets of integers with positive upper density, but there are other equivalent formulations. A “finitary” version of the theorem is as follows. Proposition 2.1 (Szemerédi’s theorem). =-=[37, 38]-=- Let N be a positive integer and let ZN := Z/NZ. 1 Let δ > 0 be a fixed positive real number, and let k � 3 be an integer. Then there is a minimal N0(δ, k) < ∞ with the following property. If N � N0(δ... |

67 |
On certain sets of positive density
- Varnavides
- 1959
(Show Context)
Citation Context ...d not just one. Once again, such a statement can be deduced from Proposition 2.1 with some combinatorial trickery (of a less trivial nature this time – the argument was first worked out by Varnavides =-=[43]-=-). A direct proof of Proposition 2.3 can be found in [40]. A formulation of Szemerédi’s theorem similar to this one was also used by Furstenberg [10]. Combining this argument with the one in Gowers gi... |

61 |
On the representation of a large even integer as the sum of a prime and the product of at most two primes.
- Chen
- 1973
(Show Context)
Citation Context ...hmetic progressions for many sets of primes for which one can give a lower bound which agrees with some upper bound coming from a sieve, up to a multiplicative constant. Invoking Chen’s famous theorem=-=[7]-=- to the effect that there are ≫ N/ log 2 N primes p � N for which p + 2 is a prime or a product of two primes, it ought to be a simple matter to adapt our arguments to show that there are arbitrarily ... |

55 | A quantitative ergodic theory proof of Szemeredi’s theorem,
- Tao
- 2006
(Show Context)
Citation Context ...ed from Proposition 2.1 with some combinatorial trickery (of a less trivial nature this time – the argument was first worked out by Varnavides [43]). A direct proof of Proposition 2.3 can be found in =-=[40]-=-. A formulation of Szemerédi’s theorem similar to this one was also used by Furstenberg [10]. Combining this argument with the one in Gowers gives an explicit bound on c(k, δ) of the form c(k, δ) � ex... |

52 | Roth’s theorem in the primes
- Green
(Show Context)
Citation Context ... statement of Theorem 1.2 by the set of all positive integers Z + , then this is a famous theorem of Szemerédi [38]. The special case k = 3 of Theorem 1.2 was recently established by the first author =-=[21]-=- using methods of Fourier analysis. In contrast, our methods here have a more ergodic theory flavour and do not involve much Fourier analysis (though the argument does rely on Szemerédi’s theorem whic... |

50 | Arithmetic progressions of length three in subsets of a random set
- Kohayakawa, Luczak, et al.
- 1996
(Show Context)
Citation Context ...We also remark that if the primes were replaced by a random subset of the integers, with density at least N −1/2+ε on each interval [1, N], then the k = 3 case of the above theorem was established in =-=[30]-=-. Acknowledgements The authors would like to thank Jean Bourgain, Enrico Bombieri, Tim Gowers, Bryna Kra, Elon Lindenstrauss, Imre Ruzsa, Roman Sasyk, Peter Sarnak and Kannan Soundararajan for helpful... |

45 | Yildirim Higher correlations of divisor sums related to primes, I: Triple correlations
- Goldston, Y
(Show Context)
Citation Context ...l be a small power of N). Define ΛR(n) := ∑ µ(d) log(R/d) = ∑ µ(d) log(R/d)+. d|n d�R These truncated divisor sums have been studied in several papers, most notably the works of Goldston and Yıldırım =-=[15, 16, 17]-=- concerning the problem of finding small gaps between primes. We shall use a modification of their arguments for obtaining asymptotics for these truncated primes to prove that the measure ν defined be... |

35 | Restriction theory of the Selberg sieve, with applications,
- Green, Tao
- 2006
(Show Context)
Citation Context ...ener algebra A in place of (U 2 ) ∗ in the k = 3 case of the arguments here. To obtain this estiamte, however, requires some serious harmonic analysis related to the restriction phenomenon (the paper =-=[23]-=- may be consulted for further information). Such a property does not seem to follow simply from the pseudorandomness of ν, and generalisation to U k−1 , k > 3, seems very difficult (it is not even cle... |

34 |
Sets of integers containing not more than a given numbers of terms in arithmetical progression,
- Rankin
- 1961
(Show Context)
Citation Context ...f of Theorem 1.2, we will need to use Szemerédi’s theorem, but we will not need any quantitative estimimates on N0(δ, k). Let us state, for contrast, the best known lower bound which is due to Rankin =-=[35]-=- (see also Lacey-̷Laba [31]): N0(δ, k) � exp(C(log 1/δ) 1+⌊log 2 (k−1)⌋ ). At the moment it is clear that a substantial new idea would be required to obtain a result of the strength (2.1). In fact, ev... |

29 |
der Corput, Über Summen von Primzahlen und Primzahlquadraten
- van
- 1939
(Show Context)
Citation Context ...blishing this conjecture here, obtaining instead a lower bound (γ(k) + o(1))N 2 / log k N for some very small γ(k) > 0). The first theoretical progress on these conjectures was made by van der Corput =-=[42]-=- (see also [8]) who, in 1939, used Vinogradov’s method of prime number sums to establish the case k = 3, that is to say that there are infinitely many triples of primes in arithmetic progression. Howe... |

23 | Pointwise convergence of ergodic averages along cubes, preprint
- Assani
(Show Context)
Citation Context ...:ω̸=0 k−1 where 0 k−1 denotes the element of {0, 1} k−1 consisting entirely of zeroes. Remark. Such functions have arisen recently in work of Host and Kra [28] in the ergodic theory setting (see also =-=[1]-=-). The next lemma, while simple, is fundamental to our entire approach; it asserts that if a function majorised by a pseudorandom measure ν is not Gowers uniform, then it correlates 13 with a bounded ... |

17 |
Convergence of Conze-Lesigne averages, Ergodic Theory Dynam
- Host, Kra
(Show Context)
Citation Context ... equations in sets of integers) and certain parts of ergodic theory. Particularly exciting is the suspicion that the notion of a k-step nilsystem, explored in many ergodic-theoretical works (see e.g. =-=[27, 28, 29, 44]-=-), might be analogous to a kind of “higher order Fourier analysis” which could be used to deal with systems of linear equations that cannot be handled by conventional Fourier analysis (a simple exampl... |

15 |
relations amongst sums of two squares, Number theory and algebraic geometry — to Peter Swinnerton-Dyer on his 75th birthday, CUP
- Linear
- 2003
(Show Context)
Citation Context ...mall k than was the case for the primes. Let S be the set of sums of two squares. It is a simple matter to show that there are infinitely many 4-term arithmetic progressions in S. Indeed, Heath-Brown =-=[26]-=- observed that the numbers (n−1) 2 +(n−8) 2 , (n−7) 2 +(n+4) 2 , (n+7) 2 +(n−4) 2 and (n+1) 2 +(n+8) 2 always form such a progression; in fact, he was able to prove much more, in particular finding an... |

12 |
A mean ergodic theorem for 1/N ∑N n=1 f(T nx)g(T n2x), Convergence in ergodic theory and probability
- Furstenberg, Weiss
- 1993
(Show Context)
Citation Context ...s of order k − 2 appear to be played by k − 2-step nilfactors (see [28, 29, 44]), which may contain polynomial eigenfunctions of order k − 2, but can also exhibit slightly more general behaviour; see =-=[14]-=- for further discussion.22 BEN GREEN AND TERENCE TAO sees that it is possible to write DF(x) = ∑ aξe 2πixξ/N + E(x), ξ∈S where |aξ| � 1 and ‖E‖L ∞ � ǫ. Also, we have |S| � ǫ−2 . Thus DF is equal to a... |

11 |
Lacey,On sets of integers not containing long arithmetic progressions
- Laba, Michael
(Show Context)
Citation Context ...eed to use Szemerédi’s theorem, but we will not need any quantitative estimimates on N0(δ, k). Let us state, for contrast, the best known lower bound which is due to Rankin [35] (see also Lacey-̷Laba =-=[31]-=-): N0(δ, k) � exp(C(log 1/δ) 1+⌊log 2 (k−1)⌋ ). At the moment it is clear that a substantial new idea would be required to obtain a result of the strength (2.1). In fact, even for k = 3 the best bound... |

11 |
Additive properties of dense subsets of sifted sequences
- Ramaré, Ruzsa
(Show Context)
Citation Context ...g a majorant to study the primes is by no means new – indeed in some sense sieve theory is precisely the study of such objects. For another use of a majorant in an additive-combinatorial setting, see =-=[33, 34]-=-. It is now timely to make a few remarks concerning the proof of Proposition 3.5. It is in the first step of the proof that our original investigations began, when we made a close examination of Gower... |

10 |
Three primes and an almost prime in arithmetic progression
- Heath-Brown
- 1981
(Show Context)
Citation Context ...ritish Columbia, Vancouver, Canada. The second author was a Clay Prize Fellow and was supported by a grant from the Packard Foundation. 12 BEN GREEN AND TERENCE TAO such result is due to Heath-Brown =-=[25]-=-. He showed that there are infinitely many 4-term progressions consisting of three primes and a number which is either prime or a product of two primes. In a somewhat different direction, let us menti... |

9 |
Linear equations in primes, Mathematika 39
- Balog
- 1992
(Show Context)
Citation Context ...ely many 4-term progressions consisting of three primes and a number which is either prime or a product of two primes. In a somewhat different direction, let us mention the beautiful results of Balog =-=[2, 3]-=-. Among other things he shows that for any m there are m distinct primes p1, . . .,pm such that all of the averages 1 2 (pi + pj) are prime. The problem of finding long arithmetic progressions in the ... |

8 |
There exists an infinity of 3—combinations of primes
- Chowla
- 1944
(Show Context)
Citation Context ...onjecture here, obtaining instead a lower bound (γ(k) + o(1))N 2 / log k N for some very small γ(k) > 0). The first theoretical progress on these conjectures was made by van der Corput [42] (see also =-=[8]-=-) who, in 1939, used Vinogradov’s method of prime number sums to establish the case k = 3, that is to say that there are infinitely many triples of primes in arithmetic progression. However, the quest... |

8 |
ergodic averages and nilmanifolds,
- Nonconventional
- 2005
(Show Context)
Citation Context ... equations in sets of integers) and certain parts of ergodic theory. Particularly exciting is the suspicion that the notion of a k-step nilsystem, explored in many ergodic-theoretical works (see e.g. =-=[27, 28, 29, 44]-=-), might be analogous to a kind of “higher order Fourier analysis” which could be used to deal with systems of linear equations that cannot be handled by conventional Fourier analysis (a simple exampl... |

8 | Twenty-two primes in arithmetic progression
- Moran, Pritchard, et al.
- 1995
(Show Context)
Citation Context ...rkus Frind, Paul Underwood, and Paul Jobling: 56211383760397 + 44546738095860k; k = 0, 1, . . ., 22. An earlier arithmetic progression of primes of length 22 was found by Moran, Pritchard and Thyssen =-=[32]-=-: 11410337850553 + 4609098694200k; k = 0, 1, . . ., 21. Our main theorem resolves the above conjecture. Theorem 1.1. The prime numbers contain infinitely many arithmetic progressions of length k for a... |

8 |
On Snirel’man’s constant, Ann
- Ramaré
- 1995
(Show Context)
Citation Context ...g a majorant to study the primes is by no means new – indeed in some sense sieve theory is precisely the study of such objects. For another use of a majorant in an additive-combinatorial setting, see =-=[33, 34]-=-. It is now timely to make a few remarks concerning the proof of Proposition 3.5. It is in the first step of the proof that our original investigations began, when we made a close examination of Gower... |

6 |
Roth’s theorem in the primes, preprint
- Green
(Show Context)
Citation Context ... statement of Theorem 1.2 by the set of all positive integers Z + , then this is a famous theorem of Szemerédi [34]. The special case k = 3 of Theorem 1.2 was recently established by the first author =-=[18]-=- using methods of Fourier analysis. In contrast, our methods here have a more ergodic theory flavour and do not involve much Fourier analysis (though the argument does rely on Szemerédi’s theorem whic... |

5 |
A Szemerédi-type theorem for sets of positive density in R k
- Bourgain
- 1986
(Show Context)
Citation Context ...a Gowers uniform component with small Fourier coefficients, and a Gowers anti-uniform component which consists of only a few Fourier coefficients (and in particular is bounded). For related ideas see =-=[5, 21, 23]-=-. Proof of Theorem 3.5 assuming Proposition 8.1. Let f, δ be as in Theorem 3.5, and let 0 < ε ≪ δ be a parameter to be chosen later. Let B be as in the above decomposition, and write fU := (1−1Ω)(f −E... |

5 |
The ergodic-theoretical proof of Szemerédi’s theorem
- Furstenberg, Katznelson, et al.
- 1982
(Show Context)
Citation Context ...rthermore we assume the bounds then we have the estimate |F(x)| � ν(x) + 1 for all x ∈ ZN ‖DF ‖L ∞ � 22k−1 −1 + o(1). (6.6) 13 This idea was inspired by the proof of the Furstenberg structure theorem =-=[10, 13]-=-; a key point in that proof being that if a system is not (relatively) weakly mixing, then it must contain a non-trivial (relatively) almost periodic function, which can then be projected out via cond... |

5 |
gaps between primes
- Small
(Show Context)
Citation Context ...n Soundararajan for helpful conversations. We are particularly indebted to Andrew Granville for drawing our attention to the work of Goldston and Yıldırım, and to Dan Goldston for making the preprint =-=[17]-=- available. We are also indebted to Bryna Kra, Jamie Radcliffe, Lior Silberman, and Mark Watkins for corrections to earlier versions of the manuscript. We are particularly indebted to the anonymous re... |

3 |
correlations of divisor sums related to primes, III: k-correlations, preprint (available at AIM preprints
- Higher
(Show Context)
Citation Context ...l be a small power of N). Define ΛR(n) := ∑ µ(d) log(R/d) = ∑ µ(d) log(R/d)+. d|n d�R These truncated divisor sums have been studied in several papers, most notably the works of Goldston and Yıldırım =-=[15, 16, 17]-=- concerning the problem of finding small gaps between primes. We shall use a modification of their arguments for obtaining asymptotics for these truncated primes to prove that the measure ν defined be... |

2 |
gaps between primes, I, preprint available at http://front.math.ucdavis.edu
- Small
(Show Context)
Citation Context ...n Soundararajan for helpful conversations. We are particularly indebted to Andrew Granville for drawing our attention to the work of Goldston and Yıldırım, and to Dan Goldston for making the preprint =-=[17]-=- available. We are also indebted to Yong-Gao Chen and his students, Bryna Kra, Jamie Radcliffe, Lior Silberman and Mark Watkins for corrections to earlier versions of the manuscript. We are particular... |

1 |
primes and an almost prime in four linear equations
- Six
- 1998
(Show Context)
Citation Context ...ely many 4-term progressions consisting of three primes and a number which is either prime or a product of two primes. In a somewhat different direction, let us mention the beautiful results of Balog =-=[2, 3]-=-. Among other things he shows that for any m there are m distinct primes p1, . . .,pm such that all of the averages 1 2 (pi + pj) are prime. The problem of finding long arithmetic progressions in the ... |