Number of directions determined by a set in ${\mathbb{F}}_q^2$ and growth in $\mathrm{Aff}({\mathbb{F}}_q)$

Abstract

We prove that a set $A$ of at most $q$ non-collinear points in the finite plane ${\mathbb{F}}_q^2$ spans at least $\approx \frac{|A|}{\sqrt{q}}$ directions: this is based on a lower bound contained in [FST13], which we prove again together with a different upper bound than the one given therein. Then, following the procedure used in [RS18], we prove a new structural theorem about slowly growing sets in $\mathrm{Aff}({\mathbb{F}}_q)$ for any finite field ${\mathbb{F}}_q$, generalizing the analogous results in [Hel15] [Mur17] [RS18] over prime fields.

12 pages