A relative Szemerédi theorem

Abstract

The celebrated Green-Tao 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 k-term arithmetic progressions in pseudorandom subsets of \({\mathbb{Z}_N}\) of density \({N^{-c_k}}\). 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.

This is a preview of subscription content, log in to check access.

References

  1. BMS

    J. Balogh, R. Morris, and W. Samotij. Independent sets in hypergraphs. J. Amer. Math. Soc., to appear.

  2. BB

    P. Bennett and T. Bohman. A note on the random greedy independent set algorithm. arXiv:1308.3732.

  3. BR09

    B. Bollobás and O. Riordan. Metrics for sparse graphs. In: Surveys in combinatorics 2009, London Math. Soc. Lecture Note Ser., vol. 365, Cambridge University Press, Cambridge, 2009, pp. 211–287.

  4. BCLSV08

    C. Borgs, J.T. Chayes, L. Lovász, V.T. Sós, and K. Vesztergombi. Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math., 219 (2008), 1801–1851.

  5. CGW89

    Chung F.R.K., Graham R.L., Wilson R.M.: Quasi-random graphs. Combinatorica 9, 345–362 (1989)

    MATH  MathSciNet  Article  Google Scholar 

  6. CCF09

    A. Coja-Oghlan, C. Cooper, and A. Frieze. An efficient sparse regularity concept. SIAM J. Discrete Math., 23 (2009/10), 2000–2034.

  7. CFZ14

    D. Conlon, J. Fox, and Y. Zhao. Extremal results in sparse pseudorandom graphs. Adv. Math., 256 (2014), 206–290.

  8. CFZ

    D. Conlon, J. Fox, and Y. Zhao. Linear forms from the Gowers uniformity norm. Unpublished companion note.

  9. CG

    D. Conlon and W.T. Gowers, Combinatorial theorems in sparse random sets. arXiv:1011.4310.

  10. CGSS14

    D. Conlon, W.T. Gowers, W. Samotij, and M. Schacht. On the KŁR conjecture in random graphs. Israel J. Math., 203 (2014), 535–580.

  11. CM12

    B. Cook and A. Magyar. Constellations in \({{\mathbb{P}}^d}\). Int. Math. Res. Not., 2012 (2012), 2794–2816.

  12. CMT

    B. Cook, A. Magyar, and T. Titichetrakun. A multidimensional Szemerédi theorem in the primes. arXiv:1306.3025.

  13. FZ

    J. Fox and Y. Zhao. A short proof of the multidimensional Szemerédi theorem in the primes. Amer. J. Math., to appear.

  14. FR02

    P. Frankl and V. Rödl. Extremal problems on set systems. Random Structures Algorithms, 20 (2002), 131–164.

  15. FK99

    Frieze A., Kannan R.: Quick approximation to matrices and applications. Combinatorica 19, 175–220 (1999)

    MATH  MathSciNet  Article  Google Scholar 

  16. FK78

    H. Furstenberg and Y. Katznelson. An ergodic Szemerédi theorem for commuting transformations. J. Analyse Math., 34 (1978), 275–291.

  17. GY03

    D.A. Goldston and C.Y. Yıldırım. Higher correlations of divisor sums related to primes. I. Triple correlations. Integers, 3 (2003), A5, 66.

  18. Gow01

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

  19. Gow07

    W.T. Gowers. Hypergraph regularity and the multidimensional Szemerédi theorem. Ann. of Math., 166 (2007), 897–946.

  20. Gow10

    W.T. Gowers. Decompositions, approximate structure, transference, and the Hahn-Banach theorem. Bull. Lond. Math. Soc., 42 (2010), 573–606. arXiv:0811.3103.

  21. GreenPC

    B. Green. Personal communication.

  22. Gre05

    B. Green. A Szemerédi-type regularity lemma in abelian groups, with applications. Geom. Funct. Anal., 15 (2005), 340–376.

  23. GT08

    B. Green and T. Tao. The primes contain arbitrarily long arithmetic progressions. Ann. of Math., 167 (2008), 481–547.

  24. GT10

    B. Green and T. Tao. Linear equations in primes. Ann. of Math., 171 (2010), 1753–1850.

  25. Koh97

    Y. Kohayakawa. Szemerédi’s regularity lemma for sparse graphs. Foundations of computational mathematics (Rio de Janeiro, 1997), Springer, Berlin, 1997, pp. 216–230.

  26. KSV09

    D. Král’, O. Serra, and L. Vena. A combinatorial proof of the removal lemma for groups. J. Combin. Theory Ser. A, 116 (2009), 971–978.

  27. KSV12

    D. Král’, O. Serra, and L. Vena. A removal lemma for systems of linear equations over finite fields. Israel J. Math. 187 (2012), 193–207.

  28. Le11

    T.H. Lê. Green-Tao theorem in function fields. Acta Arith. 147 (2011), 129–152.

  29. LS06

    L. Lovász and B. Szegedy. Limits of dense graph sequences. J. Combin. Theory Ser. B 96 (2006), 933–957.

  30. LS07

    L. Lovász and B. Szegedy. Szemerédi’s lemma for the analyst. Geom. Funct. Anal. 17 (2007), 252–270.

  31. Mat12a

    L. Matthiesen, Correlations of the divisor function. Proc. Lond. Math. Soc. 104 (2012), 827–858.

  32. Mat12b

    L. Matthiesen. Linear correlations amongst numbers represented by positive definite binary quadratic forms. Acta Arith. 154 (2012), 235–306.

  33. NRS06

    B. Nagle, V. Rödl, and M. Schacht. The counting lemma for regular k-uniform hypergraphs. Random Structures Algorithms 28 (2006), 113–179.

  34. RTTV08

    O. Reingold, L. Trevisan, M. Tulsiani, and S. Vadhan. Dense Subsets of Pseudorandom Sets. In: 49th Annual IEEE symposium on foundations of computer science, IEEE Computer Society (2008), pp. 76–85.

  35. RS04

    V. Rödl and J. Skokan. Regularity lemma for k-uniform hypergraphs. Random Structures Algorithms 25 (2004), 1–42.

  36. RS06

    V. Rödl and J. Skokan. Applications of the regularity lemma for uniform hypergraphs. Random Structures Algorithms. 28 (2006), 180–194.

  37. RS78

    I.Z. Ruzsa and E. Szemerédi. Triple systems with no six points carrying three triangles. In: Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, Colloq. Math. Soc. János Bolyai, vol. 18, North-Holland, Amsterdam (1978), pp. 939–945.

  38. ST

    D. Saxton and A. Thomason. Hypergraph containers. Invent. Math., to appear.

  39. Sch

    M. Schacht. Extremal results for random discrete structures. Submitted.

  40. Sco11

    A. Scott. Szemerédi’s regularity lemma for matrices and sparse graphs. Combin. Probab. Comput. 20 (2011), 455–466.

  41. Sha10

    A. Shapira. A proof of Green’s conjecture regarding the removal properties of sets of linear equations. J. Lond. Math. Soc. 81 (2010), 355–373.

  42. Sol03

    J. Solymosi. Note on a Generalization of Roth’s Theorem. In: Discrete and computational geometry, Algorithms Combin., vol. 25, Springer, Berlin (2003), pp. 825–827.

  43. Sol04

    J. Solymosi. A note on a question of Erdős and Graham. Combin. Probab. Comput. 13 (2004), 263–267.

  44. Sze75

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

  45. Tao

    T. Tao. A remark on Goldston-Yıldırım correlation estimates. Unpublished.

  46. Tao06a

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

  47. Tao06b

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

  48. TZ08

    T. Tao and T. Ziegler. The primes contain arbitrarily long polynomial progressions. Acta Math., 201 (2008), 213–305.

  49. TZ13

    T. Tao and T. Ziegler. A multi-dimensional Szemerédi theorem for the primes via a correspondence principle. Israel J. Math., to appear.

  50. Tow

    H. Towsner. An analytic approach to sparse hypergraphs: hypergraph removal. arXiv:1204.1884.

  51. TTV09

    L. Trevisan, M. Tulsiani, and S. Vadhan. Regularity, Boosting, and Efficiently Simulating Every High-entropy Distribution. In: 24th Annual IEEE Conference on Computational Complexity, IEEE Computer Society, 2009, pp. 126–136.

  52. Zh14

    Y. Zhao. An arithmetic transference proof of a relative Szemerédi theorem. Math. Proc. Cambridge Philos. Soc. 156 (2014), 255–261.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to David Conlon.

Additional information

The first author was supported by a Royal Society University Research Fellowship, the second author was supported by a Simons Fellowship, NSF grant DMS-1069197, by an Alfred P. Sloan Fellowship, and by an MIT NEC Corporation Fund Award, and the third author was supported by a Microsoft Research PhD Fellowship.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Conlon, D., Fox, J. & Zhao, Y. A relative Szemerédi theorem. Geom. Funct. Anal. 25, 733–762 (2015). https://doi.org/10.1007/s00039-015-0324-9

Download citation

Keywords

  • Arithmetic Progression
  • Sparse Graph
  • Discrepancy Pair
  • Regularity Lemma
  • Removal Lemma