1 5 O ct 2 01 9 Number of directions determined by a set in F 2 q and growth in Aff ( F q )

<jats:p>We prove that a set <jats:italic>A</jats:italic> of at most <jats:italic>q</jats:italic> non-collinear points in the finite plane <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {F}_{q}^{2}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula> spans more than <jats:inline-formula><jats:alternatives><jats:tex-math>$${|A|}/\!{\sqrt{q}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>A</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mspace /> <mml:msqrt> <mml:mi>q</mml:mi> </mml:msqrt> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathrm {Aff}(\mathbb {F}_{q})$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Aff</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> for any finite field <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {F}_{q}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula>, generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields.</jats:p>

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