Number of Directions Determined by a Set in $$\mathbb {F}_{q}^{2}$$ and Growth in $$\mathrm {Aff}(\mathbb {F}_{q})

• Published in: Discrete & computational geometry 2021-12, Vol.66 (4), p.1415-1428
• Main Author: Dona, Daniele
• Format: Article
• Language: English
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-021-00284-6

We prove that a set A of at most q non-collinear points in the finite plane $$\mathbb {F}_{q}^{2}$$ F q 2 spans more than $${|A|}/\!{\sqrt{q}}$$ | 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 $$\mathrm {Aff}(\mathbb {F}_{q})$$ Aff ( F q ) for any finite field  $$\mathbb {F}_{q}$$ F q , generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields.