Upper and lower bounds for rich lines in grids

TL;DR

This paper establishes new upper and lower bounds for the number of rich lines in Cartesian product point sets, disproving a previous conjecture and linking geometric configurations to additive combinatorics.

Contribution

It provides the first bounds that connect rich lines in grids with group actions and sum-product phenomena, advancing understanding in combinatorial geometry.

Findings

Disproved a conjecture of Solymosi on rich lines in grids.

Established upper bounds using asymmetric Balog-Szemeredi-Gowers theorem.

Derived lower bounds via connections to amenability and graph expansion.

Abstract

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-Szemeredi-Gowers theorem for 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 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.