Number of Directions Determined by a Set in Fq2 and Growth in Aff(Fq)

We prove that a set A of at most q non-collinear points in the finite plane Fq2 spans more than [Formula: see text] 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(Fq) for any finite field  Fq , generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields.