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

我们证明有限平面中至多q 个非共线点的集合A \(\mathbb {F}_{q}^{2}\)跨度超过\({|A|}/\!{\ sqrt{q}}\)方向：这是基于 Fancsali 等人的下限。我们用一个不同于其中给出的上界再次证明了这一点。然后，按照 Rudnev 和 Shkredov 使用的程序，我们证明了一个新的结构定理，关于在\(\mathrm {Aff}(\mathbb {F}_{q})\) 中对于任何有限域 \(\mathbb {F}_{q}\)，概括了 Helfgott、Murphy、Rudnev 和 Shkredov 在素数域上的类似结果。