Elementary methods for incidence problems in finite fields

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 $\mathrm{\mid }P\mathrm{\mid }\ge 5q$ 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.