Szemer\'edi--Trotter-type theorems in dimension 3

TL;DR

This paper establishes optimal incidence bounds between points and lines in three-dimensional space, revealing different behaviors over real, complex, and finite fields, with implications for combinatorial geometry.

Contribution

It provides the first tight Szemerédi–Trotter-type incidence bounds in three dimensions, distinguishing between real/complex and finite field cases.

Findings

Incidence bound over real/complex: mn^{1/3}

Incidence bound over finite fields: mn^{2/5}

Bounds are optimal up to a constant factor

Abstract

We estimate the number of incidences in a configuration of $m$ lines and $n$ points in dimension 3. The main term is $mn^{1/3}$ if we work over the real or complex numbers but $mn^{2/5}$ over finite fields. Both of these are optimal, aside from a multiplicative constant that is at most 5.