On the Number of Rich Lines in Truly High Dimensional Sets

Authors Zeev Dvir, Sivakanth Gopi

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Zeev Dvir
Sivakanth Gopi

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Zeev Dvir and Sivakanth Gopi. On the Number of Rich Lines in Truly High Dimensional Sets. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 584-598, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


We prove a new upper bound on the number of r-rich lines (lines with at least r points) in a 'truly' d-dimensional configuration of points v_1,...,v_n over the complex numbers. More formally, we show that, if the number of r-rich lines is significantly larger than n^2/r^d then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor r^d can be replaced with a tight r^{d+1}. If true, this would generalize the classic Szemeredi-Trotter theorem which gives a bound of n^2/r^3 on the number of r-rich lines in a planar configuration. This conjecture was shown to hold in R^3 in the seminal work of Guth and Katz and was also recently proved over R^4 (under some additional restrictions) by Solomon and Sharir. For the special case of arithmetic progressions (r collinear points that are evenly distanced) we give a bound that is tight up to lower order terms, showing that a d-dimensional grid achieves the largest number of r-term progressions. The main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not find this polynomial by interpolation. The starting observation is that the degree r-2 Veronese embedding takes r-collinear points to r linearly dependent images. Hence, each collinear r-tuple of points, gives us a dependent r-tuple of images. We then use the design-matrix method of Barak et al. to convert these 'local' linear dependencies into a global one, showing that all the images lie in a hyperplane. This then translates into a low degree polynomial vanishing on the original set.
  • Incidences
  • Combinatorial Geometry
  • Designs
  • Polynomial Method
  • Additive Combinatorics


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  1. B. Barak, Z. Dvir, A. Wigderson, and A. Yehudayoff. Fractional Sylvester-Gallai theorems. Proceedings of the National Academy of Sciences, 2012. Google Scholar
  2. B. Barak, Z. Dvir, A. Yehudayoff, and A. Wigderson. Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes. In Proceedings of the 43rd annual ACM symposium on Theory of computing, STOC '11, pages 519-528, New York, NY, USA, 2011. ACM. Google Scholar
  3. Saugata Basu and Martin Sombra. Polynomial partitioning on varieties and point-hypersurface incidences in four dimensions. arXiv preprint arXiv:1406.2144, 2014. Google Scholar
  4. Z. Dvir. Incidence Theorems and Their Applications. Foundations and Trends in Theoretical Computer Science, 6(4):257-393, 2012. Google Scholar
  5. Z. Dvir, S. Saraf, and A. Wigderson. Improved rank bounds for design matrices and a new proof of Kelly’s theorem. Forum of Mathematics, Sigma, 2, 10 2014. Google Scholar
  6. L. Guth and N. Katz. Algebraic methods in discrete analogs of the Kakeya problem. Advances in Mathematics, 225(5):2828 - 2839, 2010. Google Scholar
  7. Larry Guth and Nets Hawk Katz. On the Erdős distinct distances problem in the plane. Annals of Mathematics, 181(1):155-190, 2015. Google Scholar
  8. Marton Hablicsek and Zachary Scherr. On the number of rich lines in high dimensional real vector spaces. arXiv preprint arXiv:1412.7025, 2014. Google Scholar
  9. J. Kollar. Szemerédi-Trotter-type theorems in dimension 3. arXiv:1405.2243, 2014. Google Scholar
  10. Izabella Laba and József Solymosi. Incidence theorems for pseudoflats. Discrete & Computational Geometry, 37(2):163-174, 2007. Google Scholar
  11. M. Rudnev. On the number of incidences between planes and points in three dimensions. arXiv:1407.0426v2, 2014. Google Scholar
  12. J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, 1980. Google Scholar
  13. Micha Sharir, Adam Sheffer, and Joshua Zahl. Improved bounds for incidences between points and circles. In Proceedings of the Twenty-ninth Annual Symposium on Computational Geometry, SoCG '13, pages 97-106, New York, NY, USA, 2013. ACM. Google Scholar
  14. Micha Sharir and Noam Solomon. Incidences between points and lines in ℝ⁴. arXiv preprint arXiv:1411.0777, 2014. Google Scholar
  15. József Solymosi and VH Vu. Distinct distances in high dimensional homogeneous sets. Contemporary Mathematics, 342:259-268, 2004. Google Scholar
  16. József Solymosi and Terence Tao. An incidence theorem in higher dimensions. Discrete and Computational Geometry, 48(2):255-280, 2012. Google Scholar
  17. E. Szemerédi and W. T. Trotter. Extremal problems in discrete geometry. Combinatorica, 3(3):381-392, 1983. Google Scholar
  18. C. Toth. The Szemerédi-Trotter theorem in the complex plane. arXiv:math/0305283v4, 2003. Google Scholar
  19. Joshua Zahl. A Szemerédi-Trotter type theorem in ℝ⁴. CoRR, abs/1203.4600, 2012. Google Scholar
  20. Joshua Zahl. A note on rich lines in truly high dimensional sets. arXiv preprint arXiv:1503.01729, 2015. Google Scholar
  21. R. Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposiumon on Symbolic and Algebraic Computation, pages 216-226. Springer-Verlag, 1979. Google Scholar
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