Acta Mathematica

, Volume 201, Issue 2, pp 213–305 | Cite as

The primes contain arbitrarily long polynomial progressions

Article

Abstract

We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P 1, …, P k  ∈ Z[m] in one unknown m with P 1(0) = … = P k (0) = 0, and given any ε > 0, we show that there are infinitely many integers x and m, with \(1 \leqslant m \leqslant x^\varepsilon\), such that x + P 1(m), …, x + P k (m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case P j  = (j − 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.

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References

  1. [1]
    Alon, N., Combinatorial Nullstellensatz. Combin. Probab. Comput., 8 (1999), 7–29.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Balog, A., Pelikán, J., Pintz, J. & Szemerédi, E., Difference sets without ϰth powers. Acta Math. Hungar., 65 (1994), 165–187.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Bateman, P. T. & Horn, R. A., A heuristic asymptotic formula concerning the distribution of prime numbers. Math. Comp., 16 (1962), 363–367.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Bergelson, V., Weakly mixing PET. Ergodic Theory Dynam. Systems, 7 (1987), 337–349.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Bergelson, V., Host, B., McCutcheon, R. & Parreau, F., Aspects of uniformity in recurrence. Colloq. Math., 84/85 (2000), 549–576.MathSciNetGoogle Scholar
  6. [6]
    Bergelson, V. & Leibman, A., Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J. Amer. Math. Soc., 9 (1996), 725–753.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Deligne, P., La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math., 43 (1974), 273–307.MathSciNetCrossRefGoogle Scholar
  8. [8]
    — La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math., 52 (1980), 137–252.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Frantzikinakis, N. & Kra, B., Polynomial averages converge to the product of integrals. Israel J. Math., 148 (2005), 267–276.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Furstenberg, H., Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math., 31 (1977), 204–256.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981.MATHGoogle Scholar
  12. [12]
    Furstenberg, H. & Katznelson, Y., An ergodic Szemerédi theorem for commuting transformations. J. Anal. Math., 34 (1978), 275–291 (1979).MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Goldston, D. A., Pints, J. & Yıldırım, C. Y., Small gaps between primes. II. Preprint, 2008.Google Scholar
  14. [14]
    Goldston, D. A. & Yıldırım, C. Y., Higher correlations of divisor sums related to primes. I. Triple correlations. Integers, 3 (2003), A5, 66 pp.Google Scholar
  15. [15]
    Gowers, W. T., A new proof of Szemerédi’s theorem. Geom. Funct. Anal., 11 (2001), 465–588.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Green, B., On arithmetic structures in dense sets of integers. Duke Math. J., 114 (2002), 215–238.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Green, B. & Tao, T., An inverse theorem for the Gowers U 3 (G) norm. Proc. Edinb. Math. Soc., 51 (2008), 73–153.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    — The primes contain arbitrarily long arithmetic progressions. Ann. of Math., 167 (2008), 481–547.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    — Linear equations in primes. To appear in Ann. of Math. Google Scholar
  20. [20]
    Host, B., Progressions arithmétiques dans les nombres premiers (d’après B. Green et T. Tao). Astérisque, 307 (2006), viii, 229–246.Google Scholar
  21. [21]
    Host, B. & Kra, B., Convergence of polynomial ergodic averages. Israel J. Math., 149 (2005), 1–19.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    Janusz, G. J., Algebraic Number Fields. Pure and Applied Mathematics, 55. Academic Press, New York–London, 1973.Google Scholar
  23. [23]
    Leibman, A., Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math., 146 (2005), 303–315.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Pintz, J., Steiger, W. L. & Szemerédi, E., On sets of natural numbers whose difference set contains no squares. J. London Math. Soc., 37 (1988), 219–231.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    Ramaré, O., On Shnirel′man’s constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 645–706.MathSciNetMATHGoogle Scholar
  26. [26]
    Ramaré, O. & Ruzsa, I. Z., Additive properties of dense subsets of sifted sequences. J. Théor. Nombres Bordeaux, 13 (2001), 559–581.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    Ruzsa, I. Z., An analog of Freiman’s theorem in groups. Astérisque, 258 (1999), xv, 323–326.Google Scholar
  28. [28]
    Sárközy, A., On difference sets of sequences of integers. I. Acta Math. Acad. Sci. Hungar., 31 (1978), 125–149.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    Slijepčević, S., A polynomial Sárközy–Furstenberg theorem with upper bounds. Acta Math. Hungar., 98 (2003), 111–128.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    Szemerédi, E., On sets of integers containing no k elements in arithmetic progression. Acta Arith., 27 (1975), 199–245.MathSciNetMATHGoogle Scholar
  31. [31]
    Tao, T., The Gaussian primes contain arbitrarily shaped constellations. J. Anal. Math., 99 (2006), 109–176.MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    — Obstructions to uniformity and arithmetic patterns in the primes. Pure Appl. Math. Q., 2 (2006), 395–433.MathSciNetMATHGoogle Scholar
  33. [33]
    — A quantitative ergodic theory proof of Szemerédi’s theorem. Electron. J. Combin., 13 (2006), Research Paper 99, 49 pp.Google Scholar
  34. [34]
    — A variant of the hypergraph removal lemma. J. Combin. Theory Ser. A, 113 (2006), 1257–1280.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    — An ergodic transference theorem. Unpublished notes. http://www.math.ucla.edu/∼tao/preprints/Expository/limiting.dvi.
  36. [36]
    — A remark on Goldston–Yıldırım correlation estimates. Preprint, 2007. http://www.math.ucla.edu/∼tao/preprints/Expository/gy-corr.dvi.
  37. [37]
    Titchmarsh, E. C., The Theory of the Riemann Zeta-Function. Oxford University Press, New York, 1986.MATHGoogle Scholar
  38. [38]
    Varnavides, P., On certain sets of positive density. J. London Math. Soc., 34 (1959), 358–360.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Institut Mittag-Leffler 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, Los AngelesLos AngelesU.S.A.
  2. 2.Department of MathematicsTechnion – Israel Institute of TechnologyHaifaIsrael

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