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The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view

Author: Bryna Kra
Journal: Bull. Amer. Math. Soc. 43 (2006), 3-23
MSC (2000): Primary 11N13, 37A45, 11B25
Published electronically: October 6, 2005
MathSciNet review: 2188173
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Abstract: A long-standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only prog- ress 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.

References [Enhancements On Off] (What's this?)

  • 1. A. Balog. The prime $ k$-tuplets conjecture on average. Analytic number theory (Allerton Park, IL, 1989), 47-75, Progr. Math., 85, Birkhäuser Boston, 1990. MR 1084173 (92e:11105)
  • 2. A. Balog. Linear equations in primes. Mathematika, 39 (1992), 367-378. MR 1203292 (93m:11103)
  • 3. V. Bergelson. Weakly mixing PET. Erg. Th. & Dyn. Sys., 7 (1987), 337-349. MR 0912373 (89g:28022)
  • 4. V. Bergelson and A. Leibman. Polynomial extensions of van der Waerden's and Szemerédi's theorems. J. Amer. Math. Soc., 9 (1996), 725-753. MR 1325795 (96j:11013)
  • 5. J.-P. Conze and E. Lesigne. Sur un théorème ergodique pour des mesures diagonales. C. R. Acad. Sci. Paris, Série I, 306 (1988), 491-493. MR 0939438 (89e:22012)
  • 6. P. Erdös and P. Turán. On some sequences of integers. J. London Math. Soc., 11 (1936), 261-264.
  • 7. M. Frind, P. Jobling and P. Underwood. $ 23$ primes in arithmetic progression. Available at
  • 8. H. Furstenberg. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. d'Analyse Math., 31 (1977), 204-256. MR 0498471 (58:16583)
  • 9. H. Furstenberg and Y. Katznelson. An ergodic Szemerédi theorem for commuting transformations. J. d'Analyse Math., 34 (1979), 275-291. MR 0531279 (82c:28032)
  • 10. H. Furstenberg and Y. Katznelson. A density version of the Hales-Jewett Theorem. J. d'Analyse Math., 57 (1991), 64-119. MR 1191743 (94f:28020)
  • 11. H. Furstenberg and B. Weiss. A mean ergodic theorem for $ \frac{1}{N} \sum_{n=1}^n f({T}^{n}x)g({T}^{n^2}x)$. In Convergence in Ergodic Theory and Probability, Ohio State Univ. Math. Res. Inst. Publ., 5, Walter de Gruyter & Co, Berlin, 1996, 193-227. MR 1412607 (98e:28019)
  • 12. D. Goldston and C. Y. Yildirim. Small gaps between primes, I. Preprint.
  • 13. T. Gowers. A new proof of Szemerédi's Theorem. GAFA, 11 (2001), 465-588. MR 1844079 (2002k:11014)
  • 14. A. Granville. Personal communication.
  • 15. B. Green. Roth's Theorem in the primes. To appear in Ann. Math.
  • 16. B. Green and T. Tao. The primes contain arbitrarily long arithmetic progressions. To appear, Ann. Math.
  • 17. B. Green and T. Tao. A bound for progressions of length $ k$ in the primes. Preprint.
  • 18. G. H. Hardy and J. E. Littlewood. Some problems of ``partitio numerorum'' III: on the expression of a number as a sum of primes. Acta Math., 44 (1923), 1-70.
  • 19. D. R. Heath-Brown. Three primes and an almost prime in arithmetic progression. J. London Math. Soc. (2), 23 (1981), 396-414. MR 0616545 (82j:10074)
  • 20. B. Host and B. Kra. Convergence of Conze-Lesigne Averages. Erg. Th. & Dyn. Sys., 21 (2001), 493-509. MR 1827115 (2002d:28007)
  • 21. B. Host and B. Kra. Nonconventional ergodic averages and nilmanifolds. Ann. Math., 161 (2005), 397-488.
  • 22. B. Host and B. Kra. Convergence of polynomial ergodic averages. Israel J. Math., 149 (2005), 1-19.
  • 23. A. Leibman. Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math., 146 (2005), 303-316.
  • 24. F. Mertens. Ein Beitrag zur analytischen Zahlentheorie. Journal für Math., 78 (1874), 46-62.
  • 25. A. Moran, P. Pritchard and A. Thyssen. Twenty-two primes in arithmetic progression. Math. Comp., 64 (1995), 1337-1339. MR 1297475 (95j:11003)
  • 26. A. Selberg. The general sieve method and its place in prime number theory. Proc. ICM, vol. 1, Cambridge (1950), 286-292. MR 0044563 (13:438d)
  • 27. E. Szemerédi. On sets of integers containing no $ k$ elements in arithmetic progression. Acta Arith., 27 (1975), 199-245. MR 0369312 (51:5547)
  • 28. T. Tao. A quantitative ergodic theory proof of Szemerédi's theorem. Preprint.
  • 29. T. Tao. A remark on Goldston-Yildirim correlation estimates. Preprint.
  • 30. J. G. van der Corput. Über Summen von Primzahlen und Primzahlquadraten. Math. Ann., 116 (1939), 1-50.
  • 31. B. L. van der Waerden. Beweis einer Baudetschen Vermutung. Nieuw Arch. Wisk., 15 (1927), 212-216.
  • 32. T. Ziegler. Universal characteristic factors and Furstenberg averages. Preprint.

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Additional Information

Bryna Kra
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Rd., Evanston, Illinois 60208-2730

Received by editor(s): July 1, 2005
Published electronically: October 6, 2005
Additional Notes: This article is an expanded version of a lecture presented January 7, 2005, at the AMS Special Session on Current Events, Joint Mathematics Meetings, Atlanta, GA. The author was partially supported by NSF grant DMS-0244994.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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