Number of Directions Determined by a Set in F2q and Growth in Aff(Fq)

Abstract

We prove that a set A of at most q non-collinear points in the finite plane F2q 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(Fq) for any finite field Fq , generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields.