On the number of incidences between points and planes in three dimensions

TL;DR

This paper establishes an incidence bound between points and planes in three-dimensional projective space over various fields, extending geometric incidence estimates to positive characteristic fields and deriving bounds on distinct distances and bilinear form values.

Contribution

It introduces a new incidence theorem in $ ext{P}^3$ over arbitrary fields with characteristic not 2, generalizing Guth and Katz's approach and providing bounds in positive characteristic settings.

Findings

Incidence bound: $O(m\sqrt{n}+ m k)$ for points and planes

Bound on distinct values of bilinear forms on point sets

Lower bounds on distinct distances in finite fields

Abstract

We prove an incidence theorem for points and planes in the projective space $\mathbb P^3$ over any field $\mathbb F$, whose characteristic $p\neq 2.$ An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving $p$ if $p>0$. This yields a bound on the number of incidences between $m$ points and $n$ planes in $\mathbb P^3$, with $m\geq n$ as $$O\left(m\sqrt{n}+ m k\right),$$ where $k$ is the maximum number of collinear planes, provided that $n=O(p^2)$ if $p>0$. Examples show that this bound cannot be improved without additional assumptions. This gives one a vehicle to establish geometric incidence estimates when $p>0$. For a non-collinear point set $S\subseteq \mathbb F^2$ and a non-degenerate symmetric or skew-symmetric bilinear form $\omega$, the number of distinct values of $\omega$ on pairs of points of $S$ is $\Omega\left[\min\left(|S|^{\frac{2}{3}},p\right)\right]$. This is also the best known bound over $\mathbb R$, where it follows from the Szemer\'edi-Trotter theorem. Also, a set $S\subseteq \mathbb F^3$, not supported in a single semi-isotropic plane contains a point, from which $\Omega\left[\min\left(|S|^{\frac{1}{2}},p\right)\right]$ distinct distances to other points of $S$ are attained.