An explicit incidence theorem in F_p

Abstract

Let $P = A\times A \subset \mathbb{F}_p \times \mathbb{F}_p$, $p$ a prime. Assume that $P= A\times A$ has $n$ elements, $n<p$. See $P$ as a set of points in the plane over $\mathbb{F}_p$. We show that the pairs of points in $P$ determine $\geq c n^{1 + {1/267}}$ lines, where $c$ is an absolute constant. We derive from this an incidence theorem: the number of incidences between a set of $n$ points and a set of $n$ lines in the projective plane over $\F_p$ ($n<\sqrt{p}$) is bounded by $C n^{{3/2}-{1/10678}}$, where $C$ is an absolute constant.