On lines, joints, and incidences in three dimensions

We extend (and somewhat simplify) the algebraic proof technique of Guth and Katz [7], 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(m1/3 n) for m ≥ n, and Θ(m2/3 n2/3 +m+n) for m ≤ n. (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 proof of Bourgain’s conjecture on incidenc...

• Type: Journal
• Year: 2011
• Where: JCT
• Authors: György Elekes, Haim Kaplan, Micha Sharir
• Keywords: JCT 2011, Joints, Maximum Possible Number, Non-coplanar Lines