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Three-point configurations determined by subsets of \(\mathbb{F}_q^2\) via the Elekes-Sharir Paradigm

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Abstract

We prove that if \(E \subset \mathbb{F}_Q^2\), q ≡ 3 mod 4, has size greater than \(Cq^{\tfrac{7} {4}}\), then E determines a positive proportion of all congruence classes of triangles in \(\mathbb{F}_q^2\).

The approach in this paper is based on the approach to the Erdős distance problem in the plane due to Elekes and Sharir, followed by an incidence bound for points and lines in \(\mathbb{F}_q^3\). We also establish a weak lower bound for a related problem in the sense that any subset E of \(\mathbb{F}_q^2\) of size less than cq 4/3 definitely does not contain a positive proportion of translation classes of triangles in the plane. This result is a special case of a result established for n-simplices in \(\mathbb{F}_q^d\). Finally, a necessary and sufficient condition on the lengths of a triangle for it to exist in \(\mathbb{F}^2\) for any field \(\mathbb{F}\) of characteristic not equal to 2 is established as a special case of a result for d-simplices in \(\mathbb{F}^d\).

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References

  1. J. Bourgain: A Szemerédi type theorem for sets of positive density, Israel J. Math. 54 (1986), no. 3, 307–331.

    Article  MathSciNet  Google Scholar 

  2. J. Bourgain, N. Katz and T. Tao: A sum-product estimate in finite flelds, and applications, Geom. Func. Anal. 14 (2004), 27–57.

    Article  MATH  MathSciNet  Google Scholar 

  3. P. Brass, W. Moser and J. Pach: Research Problems in Discrete Geometry, Springer (2005).

    MATH  Google Scholar 

  4. J. Chapman, B. Erdogan, D. Hart, A. Iosevich and D. Koh: Pinned distance sets, k-simplices, Wolff’s exponent in finite flelds and sum-product estimates, Math-ematische Zeitschrift, (online; paper version to appear), (2011).

    Google Scholar 

  5. K. B. Chilakamarri: Unit-distance graphs in rational n-spaces, Discrete Math. 69 (1988), 213–218.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. Covert, D. Hart, A. Iosevich, S. Senger and I. Uriarte-Tuero: An analog of the Furstenberg-Katznelson-Weiss theorem on triangles in sets of positive density in finite fleld geometries, Discrete Math. 311 (2011), 423–430.

    Article  MATH  MathSciNet  Google Scholar 

  7. G. Elekes and M. Sharir: Incidences in three dimensions and distinct distances in the plane, Combin. Probab. Comput. 20 (2011), 571–608.

    Article  MATH  MathSciNet  Google Scholar 

  8. C. Elsholtz, W. Klotz: Maximal Dimension of Unit Simplices, Discrete and Computational Geometry, 34 (2005), 167–177.

    Article  MATH  MathSciNet  Google Scholar 

  9. P. Erdős: On sets of distances of n points, Amer. Math. Monthly 53 (1946), 248–250.

    Article  MathSciNet  Google Scholar 

  10. P. Erdős and G. Purdy: Some extremal problems in geometry, J. Combin. Theory Ser. A 10 (1971), 246–252.

    Article  MATH  Google Scholar 

  11. P. Erdős and G. Purdy: Some extremal problems in geometry III, Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), 291–308, Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg (1975).

    Google Scholar 

  12. P. Erdős and G. Purdy: Some extremal problems in geometry IV, Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State Univ., Baton Rouge, La., 1976), 307–322, Congressus Numerantium, No. XVII, Utilitas Math., Winnipeg (1976).

    Google Scholar 

  13. P. Erdős and G. Purdy: Some extremal problems in geometry V, Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977), 569–578, Congressus Numerantium, No. XIX, Utilitas Math., Winnipeg (1977).

    Google Scholar 

  14. P. Erdős and G. Purdy: Some combinatorial problems in the plane, J. Combin. Theory Ser. A 25 (1978), 205–210.

    Article  MathSciNet  Google Scholar 

  15. P. Erdős and G. Purdy: Extremal problems in combinatorial geometry, in: Handbook of Combinatorics, 2 vols, 809–874, Elsevier Sci. B. V., Amsterdam (1995).

    Google Scholar 

  16. K. J. Falconer: The geometry of fractal sets, Cambridge Tracts in Mathematics, 85 Cambridge Univ. Pr., Cambridge (1986).

    Google Scholar 

  17. H. Furstenberg, Y. Katznelson and B. Weiss: Ergodic theory and configurations in sets of positive density, Mathematics of Ramsey theory, 184–198, Algorithms Combin., 5, Springer, Berlin (1990).

    Book  Google Scholar 

  18. A. Greeneleaf and A. Iosevich: On three point configurations determined by subsets of the Euclidean plane, the associated bilinear operator and applications to discrete geometry, Analysis and PDE (accepted for publication), (2010).

    Google Scholar 

  19. L. Guth and N. Katz: On the Erdős distinct distance problem in the plane, (preprint) http://arxiv.org/pdf/1011.4105.

  20. S. Hofmann and A. Iosevich: Circular averages and Falconer/Erdős distance conjecture in the plane for random metrics, Proc. Amer. Math. Soc. 133 (2005), 133–143.

    Article  MATH  MathSciNet  Google Scholar 

  21. A. Iosevich, H. Jorati and I. Laba: Geometric incidence theorems via Fourier analysis, Trans. Amer. Math. Soc. 361 (2009), 6595–6611.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Iosevich, I. Laba: K-distance sets, Falconer conjecture, and discrete analogues, Integers 5 (2005), 11.

    MathSciNet  Google Scholar 

  23. A. Iosevich and M. Rudnev: Erdős distance problem in vector spaces over finite flelds, Transactions of the American Mathematical Society, (2007).

    Google Scholar 

  24. P. Mattila: Geometry of sets and measures in Euclidean spaces, Cambridge Univ. Pr., 44 (1995).

    Book  MATH  Google Scholar 

  25. D. Robinson: A course in the theory of groups, Second Edition, Graduate Texts in Mathematics, Springer, 80, (1995).

    MATH  Google Scholar 

  26. J. Spencer, E. Szemerédi and W. T. Trotter, Jr.: Unit distances in the Euclidean plane, Graph theory and combinatorics, 293–303, Academic Press, London, (1984).

    Google Scholar 

  27. E. Szemerédi and W. T. Trotter, Jr.: Extremal problems in discrete geometry, Combinatorica 3 (1983), 381–392.

    Article  MATH  MathSciNet  Google Scholar 

  28. L. Székely: A. Crossing numbers and hard Erdős problems in discrete geometry Combin. Probab. Comput. 6 (1997), 353–358.

    Article  MathSciNet  Google Scholar 

  29. L. Vinh: Szemeredi-Trotter type theorem and sum-product estimate in finite flelds, (preprint), http://arxiv.org/pdf/0711.4427 (2008).

    Google Scholar 

  30. T. Ziegler: Nilfactors of ℝd actions and configurations in sets of positive upper density in ℝm, J. Anal. Math. 99 (2006), 249–266.

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Mike Bennett.

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The authors were supported by NSF grant DMS-1045404.

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Bennett, M., Iosevich, A. & Pakianathan, J. Three-point configurations determined by subsets of \(\mathbb{F}_q^2\) via the Elekes-Sharir Paradigm. Combinatorica 34, 689–706 (2014). https://doi.org/10.1007/s00493-014-2978-6

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  • DOI: https://doi.org/10.1007/s00493-014-2978-6

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