# On rich points and incidences with restricted sets of lines in 3-space

Micha Sharir*, Noam Solomon

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

## 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(n4/3+ε/r2), for r ≥ 3 and for any ε > 0, and, if at most n1/3 lines of L lie on any common regulus, there are at most O(n4/3+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 [16] 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(m3/5n3/5 + (m11/15n2/5 + m1/3n2/3)s1/3 + m + n), provided that no plane contains more than s of the points. When s = O(min{n3/5/m2/5, m1/2}), the bound becomes O(m3/5n3/5 + 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(m3/5n3/5 +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.

Original language English 37th International Symposium on Computational Geometry, SoCG 2021 Kevin Buchin, Eric Colin de Verdiere Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing 9783959771849 https://doi.org/10.4230/LIPIcs.SoCG.2021.56 Published - 1 Jun 2021 37th International Symposium on Computational Geometry, SoCG 2021 - Virtual, Buffalo, United StatesDuration: 7 Jun 2021 → 11 Jun 2021

### Publication series

Name Leibniz International Proceedings in Informatics, LIPIcs 189 1868-8969

### Conference

Conference 37th International Symposium on Computational Geometry, SoCG 2021 United States Virtual, Buffalo 7/06/21 → 11/06/21

## Keywords

• Incidences
• Lines in space
• Polynomial partitioning
• Rich points

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