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 $|P|\geq 5q$ 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. 
Javier Cilleruelo, Alex Iosevich, Ben Lund, Oliver Roche-Newton and   Misha Rudnev
math.CO