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Note on the Number of Hinges Defined by a Point Set in ℝ2


It is shown that the number of distinct types of three-point hinges, defined by a real plane set of n points is ≫n2 log−3n, where a hinge is identified by fixing two pairwise distances in a point triple. This is achieved via strengthening (modulo a logn factor) of the Guth- Katz estimate for the number of pairwise intersections of lines in ℝ3, arising in the context of the plane Erdős distinct distance problem, to a second moment incidence estimate. This relies, in particular, on the generalisation of the Guth-Katz incidence bound by Solomon and Sharir.

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  1. [1]

    G. ELEKES and M. SHARIR: Incidences in three dimensions and distinct distances in the plane, Combin. Probab. Corn-put.20 (2011), 571–608.

    MathSciNet  Article  Google Scholar 

  2. [2]

    P. ERDŐS: On sets of distances of n points, Amer. Math. Monthly53 (1946), 248–250.

    MathSciNet  Article  Google Scholar 

  3. [3]

    L. GUTH and N. H. KATZ: On the Erdos distinct distances problem in the plane, Ann. of Math. (2) 181 (2015), 155–190.

    MathSciNet  Article  Google Scholar 

  4. [4]

    A. IOSEVICH and H. PARSHALL: Embedding distance graphs in finite field vector spaces, J. Korean Math. Soc.56 (2019), 1515–1528.

    MathSciNet  MATH  Google Scholar 

  5. [5]

    A. IOSEVICH and J. PASSANT: Finite point configurations in the plane, rigidity and Erdos problems, Proc. Steklov Math. Inst.303 (2018), 129–139.

    Article  Google Scholar 

  6. [6]

    E. A. PALSSON, S. SENGER and A. SHEFFER: On the Number of Discrete Chains, arXiv:1902.08259 [math.CO] 21 Feb 2019.

    Google Scholar 

  7. [7]

    M. RUDNEV: On the number of classes of triangles determined by N points in R2, arXiv:1205.4865 [math.CO] 26 May 2012.

    Google Scholar 

  8. [8]

    M. RUDNEV: On the number of incidences between points and planes in three dimensions, Combinatorial38 (2018), 219–254.

    MathSciNet  Article  Google Scholar 

  9. [9]

    M. RLJDNEV and J. M. SELIG: On the use of the Klein quadric for geometric incidence problems in two dimensions, SI AM J. DISCRETE Math.30 (2016), 934–954.

    MathSciNet  Article  Google Scholar 

  10. [10]

    M. SHARIR and N. SOLOMON: Incidences between points and lines on two- and three-dimensional varieties, Discrete Comput. Geom.59 (2018), 88–130.

    MathSciNet  Article  Google Scholar 

  11. [11]

    F. de ZEEUW: A short proof of Rudnev’s point-plane incidence bound, arXiv:1612.02719vl [math.CO] 8 Dec 2016.

    Google Scholar 

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Correspondence to Misha Rudnev.

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Partially supported by the Leverhulme Trust Grant RPG-2017-371.

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Rudnev, M. Note on the Number of Hinges Defined by a Point Set in ℝ2. Combinatorica 40, 749–757 (2020).

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