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 $\ge c{n}^{1+1/267}$ 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.

11 pages

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