Upper and lower bounds for rich lines in grids

We prove upper and lower bounds for the number of lines in general position that are rich in a Cartesian product point set. This disproves a conjecture of Solymosi and improves work of Elekes, Borenstein and Croot, and Amirkhanyan, Bush, Croot, and Pryby.

The upper bounds are based on a version of the asymmetric Balog-Szemer\'{e}di-Gowers theorem for {\it group actions} combined with product theorems for the affine group. The lower bounds are based on a connection between rich lines in Cartesian product sets and {\it amenability} (or expanding families of graphs in the finite field case).

As an application of our upper bounds for rich lines in grids, we give a geometric proof of the asymmetric sum-product estimates of Bourgain and Shkredov.