Number of Directions Determined by a Set in F q 2 and Growth in Aff ( F q ).

We prove that a set A of at most q non-collinear points in the finite plane F q 2 spans more than | A | / q directions: this is based on a lower bound by Fancsali et al. which we prove again together with a different upper bound than the one given therein. Then, following the procedure used by Rudnev and Shkredov, we prove a new structural theorem about slowly growing sets in Aff ( F q ) for any finite field  F q , generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields.