Elementary methods for incidence problems in finite fields

Abstract

We use elementary methods to prove an incidence theorem for points and spheres in ${\mathbb{F}}_q^n$. As an application, we show that any point set of $P\subset {\mathbb{F}}_q^2$ with $|P|\ge 5q$ determines a positive proportion of all circles. The latter result is an analogue of Beck's Theorem for circles which is optimal up to multiplicative constants.

9 pages. In this new version, Theorem 3 has been significantly improved, whilst the proof has been simplified. Also, Ben Lund has been added as an author