An Energy Bound in the Affine Group
Combinatorics
2022-05-05 v2
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.
Cite
@article{arxiv.1911.03401,
title = {An Energy Bound in the Affine Group},
author = {Giorgis Petridis and Oliver Roche-Newton and Misha Rudnev and Audie Warren},
journal= {arXiv preprint arXiv:1911.03401},
year = {2022}
}
Comments
16 pages, 1 figure