An Energy Bound in the Affine Group

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.Comment: 16 pages, 1 figur

A Point-Conic Incidence Bound and Applications over Fp\mathbb F_p

In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These include new lower bounds on the number of pinned algebraic distances as well as improvements of results of Koh and Sun (2014) and Shparlinski (2006) on the size of the distance set formed by two large subsets of finite dimensional vector spaces over finite fields. We also prove a variant of Beck's theorem for conics.Comment: To appear in European Journal of Combinatoric