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arXiv · 1910.06752

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

Daniele Dona
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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.

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Daniele Dona. 2019-10-15. Number of directions determined by a set in $\mathbb{F}_{q}^{2}$ and growth in $\mathrm{Aff}(\mathbb{F}_{q})$. https://doi.org/10.1007/s00454-021-00284-6

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