We prove a nontrivial energy bound for a finite set of affine transformations over a general field and discuss a number of implications. These include new bounds on growth in the affine group, a quantitative version of a theorem by Elekes about rich lines in grids. We also give a positive answer to a question of Yufei Zhao that for a plane point set P for which no line contains a positive proportion of points from P, there may be at most one line, meeting the set of lines defined by P in at most a constant multiple of |P| points.

中文翻译：

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仿射群中的能量界

我们证明了一般域上有限仿射变换集的非平凡能量界，并讨论了许多含义。其中包括仿射群中增长的新边界，这是 Elekes 关于网格中丰富线的定理的定量版本。我们还对赵宇飞的一个问题给出了肯定的答案，即对于没有直线包含来自 P 的点的正比例的平面点集 P，最多可能有一条直线，满足 P in 定义的直线集合最大为 |P| 的常数倍 点。