Incidences between points and lines in three dimensions

We give a fairly elementary and simple proof that shows that the number of incidences between m points and n lines in R³, so that no plane contains more than s lines, is O (m^1/2n^3/4 + m^2/3n^1/3s^1/3 + m + n) (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 (Ann Math 181:155–190, 2015, [10]) as a major step in their solution of Erdős’s distinct distances problem, is also a major new result in incidence geometry, an area that has picked up considerable momentum in the past decade. 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 (Discrete Comput Geom 53(2):428–444, 2015, [8]). The present paper presents a different and simpler derivation, with better bounds than those in Guth, and without the restrictive assumptions made there. Our result has a potential for applications to other incidence problems in higher dimensions.