Document # On Rich Points and Incidences with Restricted Sets of Lines in 3-Space

### Authors Micha Sharir , Noam Solomon ## File

LIPIcs.SoCG.2021.56.pdf
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• 14 pages

## Acknowledgements

We deeply thank Larry Guth for helpful interaction on this work.

## Cite As

Micha Sharir and Noam Solomon. On Rich Points and Incidences with Restricted Sets of Lines in 3-Space. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 56:1-56:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.SoCG.2021.56

## Abstract

Let L be a set of n lines in ℝ³ that is contained, when represented as points in the four-dimensional Plücker space of lines in ℝ³, in an irreducible variety T of constant degree which is non-degenerate with respect to L (see below). We show: (1) If T is two-dimensional, the number of r-rich points (points incident to at least r lines of L) is O(n^{4/3+ε}/r²), for r ⩾ 3 and for any ε > 0, and, if at most n^{1/3} lines of L lie on any common regulus, there are at most O(n^{4/3+ε}) 2-rich points. For r larger than some sufficiently large constant, the number of r-rich points is also O(n/r). As an application, we deduce (with an ε-loss in the exponent) the bound obtained by Pach and de Zeeuw [J. Pach and F. de Zeeuw, 2017] on the number of distinct distances determined by n points on an irreducible algebraic curve of constant degree in the plane that is not a line nor a circle. (2) If T is two-dimensional, the number of incidences between L and a set of m points in ℝ³ is O(m+n). (3) If T is three-dimensional and nonlinear, the number of incidences between L and a set of m points in ℝ³ is O (m^{3/5}n^{3/5} + (m^{11/15}n^{2/5} + m^{1/3}n^{2/3})s^{1/3} + m + n), provided that no plane contains more than s of the points. When s = O(min{n^{3/5}/m^{2/5}, m^{1/2}}), the bound becomes O(m^{3/5}n^{3/5}+m+n). As an application, we prove that the number of incidences between m points and n lines in ℝ⁴ contained in a quadratic hypersurface (which does not contain a hyperplane) is O(m^{3/5}n^{3/5} + m + n). The proofs use, in addition to various tools from algebraic geometry, recent bounds on the number of incidences between points and algebraic curves in the plane.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• Lines in space
• Rich points
• Polynomial partitioning
• Incidences

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## References

1. P. Agarwal, E. Nevo, J. Pach, R. Pinchasi, M. Sharir, and S. Smorodinsky. Lenses in arrangements of pseudocircles and their applications. J. ACM, 51:139-186, 2004. 2. B. Aronov and M. Sharir. Cutting circles into pseudo-segments and improved bounds for incidences. Discrete Comput. Geom., 28:475-490, 2002. 3. J. Cardinal, M. Payne, and N. Solomon. Ramsey-type theorems for lines in 3-space. Discrete Math. Theoret. Comput. Sci., 18, 2016. 4. K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, and E. Welzl. Combinatorial complexity bounds for arrangements of curves and spheres. Discrete Comput. Geom., 5:99-160, 1990. 5. G. Elekes, H. Kaplan, and M. Sharir. On lines, joints, and incidences in three dimensions. J. Combinat. Theory, Ser. A, 118:962-977, 2011. Also in URL: https://arxiv.org/abs/0905.1583.
6. Gy. Elekes and M. Sharir. Incidences in three dimensions and distinct distances in the plane. Combinat. Probab. Comput., 20:571-608, 2011. Also in URL: https://arxiv.org/abs/1005.0982.
7. D. Fuchs and S. Tabachnikov. Mathematical Omnibus: Thirty Lectures on Classic Mathematics. Amer. Math. Soc. Press, Providence, RI, 2007. 8. W. Fulton. Introduction to Intersection Theory in Algebraic Geometry. Expository Lectures from the CBMS Regional Conference Held at George Mason University, June 27-July 1, 1983, Vol. 54. AMS Bookstore, 1984. 9. P. Griffiths and J. Harris. Principles of Algebraic Geometry, volume 52. John Wiley & Sons, New York, 2011. 10. L. Guth. Polynomial partitioning for a set of varieties. Math. Proc. Cambridge Phil. Soc., 159:459-469, 2015. Also in URL: https://arxiv.org/abs/1410.8871.
11. L. Guth and N.~H. Katz. On the ErdH os distinct distances problem in the plane. Annals Math., 181:155-190, 2015. Also in URL: https://arxiv.org/abs/1011.4105.
12. L. Guth and J. Zahl. Algebraic curves, rich points, and doubly-ruled surfaces. Amer. J. Math., 140:1187-1229, 2018. Also in URL: https://arxiv.org/abs/1503.02173.
13. J. Harris. Algebraic Geometry: A First Course, volume 133. Springer-Verlag, New York, 1992. 14. H. Kaplan, M. Sharir, and E. Shustin. On lines and joints. Discrete Comput. Geom., 44:838-843, 2010. Also in URL: https://arxiv.org/abs/0906.0558.
15. A. Marcus and G. Tardos. Intersection reverse sequences and geometric applications. J. Combinat. Theory, Ser. A, 113:675-691, 2006. 16. J. Pach and F. de Zeeuw. Distinct distances on algebraic curves in the plane. Combinat. Probab. Comput., 26:99-117, 2017. 17. J. Pach and M. Sharir. On the number of incidences between points and curves. Combinat. Probab. Comput., 7:121-127, 1998. 18. M. Rudnev. On the number of incidences between points and planes in three dimensions. Combinatorica, 38:219-254, 2018. 19. M. Sharir, A. Sheffer, and N. Solomon. Incidences with curves in R^d. Electronic J. Combinat., 23(4), 2016. Also in Proc. European Sympos. Algorithms (2015), Springer LNCS 9294, pp. 977-988, and in URL: https://arxiv.org/abs/1512.08267.
20. M. Sharir, A. Sheffer, and J. Zahl. Improved bounds for incidences between points and circles. Combinat. Probab. Comput., 24:490-520, 2015. Also in URL: https://arxiv.org/abs/1208.0053.
21. M. Sharir and N. Solomon. On rich points and incidences with restricted sets of lines in 3-space. URL: http://arxiv.org/abs/2012.11913.
22. M. Sharir and N. Solomon. Incidences between points and lines in R⁴. Discrete Comput. Geom., 57(3):702-756, 2017. Also in Proc. 56th IEEE Sympos. Foundations of Computer Science 2015, 1378-1394, and in URL: https://arxiv.org/abs/1411.0777.
23. M. Sharir and N. Solomon. Incidences between points and lines on two- and three- dimensional varieties. Discrete Comput. Geom., 59:88-130, 2018. Also in URL: https://arxiv.org/abs/1609.09026.
24. M. Sharir, N. Solomon, and O. Zlydenko. Incidences with curves with almost two degrees of freedom. https://arxiv.org/abs/2003.02190. Also in Proc. 36th Sympos. on Computational Geometry (2020), 66:1-66:14.
25. M. Sharir and J. Zahl. Cutting algebraic curves into pseudo-segments and applications. J. Combinat. Theory Ser. A, 150:1-35, 2017. 26. N. Solomon and R. Zhang. Highly incidental patterns on a quadratic hypersurface in R⁴. Discrete Math., 340:585-590, 2017. 27. L. Székely. Crossing numbers and hard ErdH os problems in discrete geometry. Combinat. Probab. Comput., 6:353-358, 1997. 28. E. Szemerédi and W.T. Trotter. Extremal problems in discrete geometry. Combinatorica, 3:381-392, 1983. 29. J. Zahl. Breaking the 3/2 barrier for unit distances in three dimensions, 2017. URL: http://arxiv.org/abs/1706.05118.
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