On lines, joints, and incidences in three dimensions

György Elekes, Haim Kaplan, Micha Sharir

Research output: Contribution to journalArticlepeer-review

Abstract

We extend (and somewhat simplify) the algebraic proof technique of Guth and Katz (2010) [9], to obtain several sharp bounds on the number of incidences between lines and points in three dimensions. Specifically, we show: (i) The maximum possible number of incidences between n lines in R3 and m of their joints (points incident to at least three non-coplanar lines) is Θ(m1/3n) for mΘn, and Θ(m2/3n2/3+m+n) for mΘn. (ii) In particular, the number of such incidences cannot exceed O(n3/2). (iii) The bound in (i) also holds for incidences between n lines and m arbitrary points (not necessarily joints), provided that no plane contains more than O(n) points and each point is incident to at least three lines. As a preliminary step, we give a simpler proof of (an extension of) the bound O(n3/2), established by Guth and Katz, on the number of joints in a set of n lines in R3. We also present some further extensions of these bounds, and give a trivial proof of Bourgain's conjecture on incidences between points and lines in 3-space, which is an immediate consequence of our incidence bounds, and which constitutes a much simpler alternative to the proof of Guth and Katz (2010) [9].

Original languageEnglish
Pages (from-to)962-977
Number of pages16
JournalJournal of Combinatorial Theory - Series A
Volume118
Issue number3
DOIs
StatePublished - Apr 2011

Keywords

  • Algebraic techniques
  • Incidences
  • Joints
  • Lines in 3-space
  • Polynomials

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