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

TL;DR

This paper establishes a lower bound on the number of directions determined by a set of points in a finite plane and introduces a structural theorem for slowly growing sets in the affine group over finite fields, generalizing previous results.

Contribution

It provides a new lower bound on directions spanned by point sets and a generalized structural theorem for slowly growing sets in affine groups over finite fields.

Findings

Set of at most q non-collinear points spans about |A|/√q directions.

Proved a new lower bound on directions based on previous bounds.

Established a structural theorem for slowly growing sets in Aff(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.