The problem of “rich lines in grids” lies at the intersection of arithmetic and geometric combinatorics: Given a point set \(P=A\times A\) in the plane, if \(L\) is a set of lines such that each line in \(L\) contains a near maximal number of points of \(P\), must \(L\) has some special structure? Although this problem seems geometric, results of this type imply non-trivial sum-product bounds, and we will give a proof of such a structure theorem using the theory of “approximate groups”. Some background in additive combinatorics would be helpful, but not essential, to understanding the talk.

If time permits, we will discuss some related problems in geometric measure theory.