A Szemer\'edi-Trotter Theorem in Arbitrary Fields

Abstract

Let $k$ be a field of characteristic $p\ge0$. We prove that $m$ points and $n$ lines in $k^2$ determine at most $3(mn)^{2/3}+m+n+2mn/p$ incidences, the last term being omitted in characteristic zero. Over the prime field $\mathbb{F}_p$ the coefficient of $mn/p$ can be replaced by $1$. The proof uses the polynomial method, and for $m=n$ the bound is sharp up to an absolute constant over prime fields. As applications, over prime fields in which $-1$ is not a square we obtain the $L^2\to L^r$ extension estimate for the paraboloid in $\mathbb{F}_p^3$ for $r>10/3$. Over every odd prime field, we show that a two-source extractor construction of Bourgain has exponentially small error at every min-entropy rate greater than $1/3$. We also improve sum-product estimates for small sets in positive characteristic and obtain projection and Furstenberg estimates over prime fields. The incidence inequalities with exact constants have been formalized in Lean.