On Lines, Joints, and Incidences in Three Dimensions

Elekes, György; Kaplan, Haim; Sharir, Micha

On Lines, Joints, and Incidences in Three Dimensions

Authors

Elekes, György

Kaplan, Haim

Sharir, Micha

Description

We extend (and somewhat simplify) the algebraic proof technique of Guth and Katz \cite{GK}, 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 $\reals^3$ and $m$ of their joints (points incident to at least three non-coplanar lines) is $Θ(m^{1/3}n)$ for $m\ge n$, and $Θ(m^{2/3}n^{2/3}+m+n)$ for $m\le n$. (ii) In particular, the number of such incidences cannot exceed $O(n^{3/2})$. (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 $O(n^{3/2})$, established by Guth and Katz, on the number of joints in a set of $n$ lines in $\reals^3$. We also present some further extensions of these bounds, and give a proof of Bourgain's conjecture on incidences between points and lines in 3-space, which constitutes a simpler alternative to the proof of \cite{GK}.