Sum-product phenomena for algebraic groups and uniformity

The classical sum-product phenomena refers to the fact that for any finite set of natural numbers, either its sum set or its product set is large. Erdös--Szemerédi conjectured a sharp lower bound for the maximum of the two. This conjecture is still open but various weaker versions have been shown. Bays--Breuillard generalized this phenomenon to algebraic groups. Further generalizations have been proved by Chernikov--Peterzil--Starchenko. Both of those groups used a mixture of model theory and incidence geometry. In joint work with Harrison and Mudgal we prove a Bourgain--Chang type result for complex algebraic groups of dimension 1. We use substantially different methods than the previous groups. Time permitting, I will also talk about applications of our methods to a question of Bremner.