Elementary methods for incidence problems in finite fields

Javier Cilleruelo, Alex Iosevich, Ben Lund, Oliver Roche-Newton, Misha Rudnev

Published 2014-07-09, updated 2014-08-17Version 2

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 $|P|\geq 5q$ 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.