Incidences Between Points and Lines in Three Dimensions

Micha Sharir,Noam Solomon

doi:10.1007/978-3-662-57413-3_15

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.

Highlights

Let P be a set of m distinct points in R3 and let L be a set of n distinct lines in R3
In the 2010 groundbreaking paper of Guth and Katz [8], an improved bound has been derived for I(P, L), for a set P of m points and a set L of n lines in R3, provided that not too many lines of L lie in a common plane
We provide a simple derivation of this bound, which bypasses most of the techniques from algebraic geometry that are used in the original proof

Summary

Introduction

Let P be a set of m distinct points in R3 and let L be a set of n distinct lines in R3. In the 2010 groundbreaking paper of Guth and Katz [8], an improved bound has been derived for I(P, L), for a set P of m points and a set L of n lines in R3, provided that not too many lines of L lie in a common plane This bound was a major step in the derivation of the main result of [8], which was to prove an almost-linear lower bound on the number of distinct distances determined by any finite set of points in the plane, a classical problem posed by Erdős in 1946 [5] Their proof uses several nontrivial tools from algebraic and differential geometry, most notably the Cayley–Salmon theorem on osculating lines to algebraic surfaces in R3, and additional properties of ruled surfaces. Where B is an absolute constant, and, for another suitable absolute constant b > 1, log(m2 n)

We skip over certain subtleties in their bound

Background

Proof of Theorem 2

Discussion