Number of Directions Determined by a Set in [Formula: see text] and Growth in [Formula: see text]

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