由$\mathbb{F}_{q}^{2}$中的集合和$\mathrm{Aff}(\mathbb{F}_{q})$中的增长确定的方向数
Number of directions determined by a set in $\mathbb{F}_{q}^{2}$ and growth in $\mathrm{Aff}(\mathbb{F}_{q})$
摘要：
我们证明了一组$A$的最多$q$非共线点在有限平面$\mathbb{F}_{q}^{2}$跨度至少$\approx\frac{|A|}{\sqrt{q}}$方向：这是基于一个下界包含在[FST 13]，我们再次证明了一个不同的上限比其中给出的。然后，按照[RS 18]中的方法，对任意有限域$\mathbb{F}_{q}$，证明了$\mathrm{Aff}(\mathbb{F}_{q})$中慢增长集的一个新的结构定理，推广了[Hel15] [Mur 17] [RS 18]中关于素域的类似结果.
英文摘要：
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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