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The primes contain arbitrarily long polynomial progressions

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Acta Mathematica

An Erratum to this article was published on 25 June 2013

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. Alon, N., Combinatorial Nullstellensatz. Combin. Probab. Comput., 8 (1999), 7–29.

    Article  MathSciNet  MATH  Google Scholar 

  2. Balog, A., Pelikán, J., Pintz, J. & Szemerédi, E., Difference sets without ϰth powers. Acta Math. Hungar., 65 (1994), 165–187.

    Article  MathSciNet  MATH  Google Scholar 

  3. Bateman, P. T. & Horn, R. A., A heuristic asymptotic formula concerning the distribution of prime numbers. Math. Comp., 16 (1962), 363–367.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bergelson, V., Weakly mixing PET. Ergodic Theory Dynam. Systems, 7 (1987), 337–349.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bergelson, V., Host, B., McCutcheon, R. & Parreau, F., Aspects of uniformity in recurrence. Colloq. Math., 84/85 (2000), 549–576.

    MathSciNet  Google Scholar 

  6. Bergelson, V. & Leibman, A., Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J. Amer. Math. Soc., 9 (1996), 725–753.

    Article  MathSciNet  MATH  Google Scholar 

  7. Deligne, P., La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math., 43 (1974), 273–307.

    Article  MathSciNet  Google Scholar 

  8. — La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math., 52 (1980), 137–252.

    Article  MathSciNet  MATH  Google Scholar 

  9. Frantzikinakis, N. & Kra, B., Polynomial averages converge to the product of integrals. Israel J. Math., 148 (2005), 267–276.

    Article  MathSciNet  MATH  Google Scholar 

  10. Furstenberg, H., Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math., 31 (1977), 204–256.

    Article  MathSciNet  MATH  Google Scholar 

  11. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981.

    MATH  Google Scholar 

  12. Furstenberg, H. & Katznelson, Y., An ergodic Szemerédi theorem for commuting transformations. J. Anal. Math., 34 (1978), 275–291 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  13. Goldston, D. A., Pints, J. & Yıldırım, C. Y., Small gaps between primes. II. Preprint, 2008.

  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. Gowers, W. T., A new proof of Szemerédi’s theorem. Geom. Funct. Anal., 11 (2001), 465–588.

    Article  MathSciNet  MATH  Google Scholar 

  16. Green, B., On arithmetic structures in dense sets of integers. Duke Math. J., 114 (2002), 215–238.

    Article  MathSciNet  MATH  Google Scholar 

  17. Green, B. & Tao, T., An inverse theorem for the Gowers U 3 (G) norm. Proc. Edinb. Math. Soc., 51 (2008), 73–153.

    Article  MathSciNet  MATH  Google Scholar 

  18. — The primes contain arbitrarily long arithmetic progressions. Ann. of Math., 167 (2008), 481–547.

    Article  MathSciNet  MATH  Google Scholar 

  19. — Linear equations in primes. To appear in Ann. of Math.

  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. Host, B. & Kra, B., Convergence of polynomial ergodic averages. Israel J. Math., 149 (2005), 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  22. Janusz, G. J., Algebraic Number Fields. Pure and Applied Mathematics, 55. Academic Press, New York–London, 1973.

    Google Scholar 

  23. Leibman, A., Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math., 146 (2005), 303–315.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  25. Ramaré, O., On Shnirel′man’s constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 645–706.

    MathSciNet  MATH  Google Scholar 

  26. Ramaré, O. & Ruzsa, I. Z., Additive properties of dense subsets of sifted sequences. J. Théor. Nombres Bordeaux, 13 (2001), 559–581.

    Article  MathSciNet  MATH  Google Scholar 

  27. Ruzsa, I. Z., An analog of Freiman’s theorem in groups. Astérisque, 258 (1999), xv, 323–326.

    Google Scholar 

  28. Sárközy, A., On difference sets of sequences of integers. I. Acta Math. Acad. Sci. Hungar., 31 (1978), 125–149.

    Article  MathSciNet  MATH  Google Scholar 

  29. Slijepčević, S., A polynomial Sárközy–Furstenberg theorem with upper bounds. Acta Math. Hungar., 98 (2003), 111–128.

    Article  MathSciNet  MATH  Google Scholar 

  30. Szemerédi, E., On sets of integers containing no k elements in arithmetic progression. Acta Arith., 27 (1975), 199–245.

    MathSciNet  MATH  Google Scholar 

  31. Tao, T., The Gaussian primes contain arbitrarily shaped constellations. J. Anal. Math., 99 (2006), 109–176.

    Article  MathSciNet  MATH  Google Scholar 

  32. — Obstructions to uniformity and arithmetic patterns in the primes. Pure Appl. Math. Q., 2 (2006), 395–433.

    MathSciNet  MATH  Google Scholar 

  33. — A quantitative ergodic theory proof of Szemerédi’s theorem. Electron. J. Combin., 13 (2006), Research Paper 99, 49 pp.

  34. — A variant of the hypergraph removal lemma. J. Combin. Theory Ser. A, 113 (2006), 1257–1280.

    Article  MathSciNet  MATH  Google Scholar 

  35. — An ergodic transference theorem. Unpublished notes. http://www.math.ucla.edu/∼tao/preprints/Expository/limiting.dvi.

  36. — A remark on Goldston–Yıldırım correlation estimates. Preprint, 2007. http://www.math.ucla.edu/∼tao/preprints/Expository/gy-corr.dvi.

  37. Titchmarsh, E. C., The Theory of the Riemann Zeta-Function. Oxford University Press, New York, 1986.

    MATH  Google Scholar 

  38. Varnavides, P., On certain sets of positive density. J. London Math. Soc., 34 (1959), 358–360.

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Correspondence to Terence Tao.

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The second author was partially supported by NSF grant DMS-0111298. This work was initiated at a workshop held at the CRM in Montreal. The authors would like to thank the CRM for their hospitality.

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Tao, T., Ziegler, T. The primes contain arbitrarily long polynomial progressions. Acta Math 201, 213–305 (2008). https://doi.org/10.1007/s11511-008-0032-5

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  • DOI: https://doi.org/10.1007/s11511-008-0032-5

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