An Energy Bound in the Affine Group

TL;DR

This paper establishes a new energy bound for finite sets of affine transformations over general fields, leading to improved growth bounds in affine groups and a quantitative version of Elekes' theorem on rich lines in grids.

Contribution

It introduces a novel energy bound for affine transformations, providing new insights into growth in affine groups and answering a question by Yufei Zhao about line incidences in point sets.

Findings

New energy bounds for affine transformations over general fields

Enhanced bounds on growth in the affine group

A positive answer to Zhao's question on line incidences in point sets

Abstract

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.