Incidences between points and lines in three dimensions

TL;DR

This paper presents a simplified proof for the incidence bound between points and lines in three-dimensional space, improving understanding and potentially aiding solutions to higher-dimensional incidence problems.

Contribution

It offers a new, simpler derivation of a key incidence bound in 3D that improves upon previous bounds and removes restrictive assumptions.

Findings

Established a bound of O(m^{1/2}n^{3/4} + m^{2/3}n^{1/3}s^{1/3} + m + n) for incidences in 3D.

Provided a more accessible proof compared to previous algebraic geometry methods.

Enhanced potential for applying incidence bounds to higher-dimensional problems.

Abstract

We give a fairly elementary and simple proof that shows that the number of incidences between $m$ points and $n$ lines in ${\mathbb R}^3$, so that no plane contains more than $s$ lines, is $$ O\left(m^{1/2}n^{3/4}+ m^{2/3}n^{1/3}s^{1/3} + m + n\right) $$ (in the precise statement, the constant of proportionality of the first and third terms depends, in a rather weak manner, on the relation between $m$ and $n$). This bound, originally obtained by Guth and Katz~\cite{GK2} as a major step in their solution of Erd{\H o}s's distinct distances problem, is also a major new result in incidence geometry, an area that has picked up considerable momentum in the past six years. Its original proof uses fairly involved machinery from algebraic and differential geometry, so it is highly desirable to simplify the proof, in the interest of better understanding the geometric structure of the problem, and providing new tools for tackling similar problems. This has recently been undertaken by Guth~\cite{Gu14}. The present paper presents a different and simpler derivation, with better bounds than those in \cite{Gu14}, and without the restrictive assumptions made there. Our result has a potential for applications to other incidence problems in higher dimensions.