An Improved Point-Line Incidence Bound Over Arbitrary Fields

TL;DR

This paper establishes a new, tighter upper bound on point-line incidences over arbitrary fields, improving previous results and applying the bound to various combinatorial geometry problems.

Contribution

It introduces an improved incidence bound over arbitrary fields, utilizing a reduction to point-plane incidences and covering techniques, advancing the understanding of geometric incidences.

Findings

New upper bound: O(m^{11/15} n^{11/15}) incidences

Bound applies to fields with positive characteristic under certain conditions

Applications to sum-product problems, expanders, and distance problems

Abstract

We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field $\mathbb{F}$, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that $m$ points and $n$ lines in $\mathbb{F}^2$, with $m^{7/8}<n<m^{8/7}$, determine at most $O(m^{11/15}n^{11/15})$ incidences (where, if $\mathbb{F}$ has positive characteristic $p$, we assume $m^{-2}n^{13}\ll p^{15}$). This improves on the previous best known bound, due to Jones. To obtain our bound, we first prove an optimal point-line incidence bound on Cartesian products, using a reduction to a point-plane incidence bound of Rudnev. We then cover most of the point set with Cartesian products, and we bound the incidences on each product separately, using the bound just mentioned. We give several applications, to sum-product-type problems, an expander problem of Bourgain, the distinct distance problem and Beck's theorem.