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arXiv · 1911.03401

An Energy Bound in the Affine Group

Giorgis Petridis·Oliver Roche-Newton·Misha Rudnev·Audie Warren
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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.

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Giorgis Petridis, Oliver Roche-Newton, Misha Rudnev, Audie Warren. 2020-05-06. An Energy Bound in the Affine Group. https://doi.org/10.1093/imrn%2Frnaa130

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