Elekes-Szabó, weak general position, and generic nilprogressions
By the Elekes-Szabó theorem, any ternary algebraic relation in characteristic 0 which has asymptotically large intersections with products of finite sets in "general position" must be, up to finite correspondences, the graph of addition in an abelian algebraic group. I will discuss some first steps towards understanding what happens when we try to relax this general position hypothesis, which in model theoretic terms is a kind of minimality. I will concentrate on the case that we already know the relation to be the graph of multiplication in an algebraic group, where (via Balog-Szemerédi-Gowers-Tao) one is really talking about the existence of approximate subgroups in some sort of general position. Our main result is that for a certain weak notion of general position, this characterises nilpotence of the group. The proof involves a generic Mordell-Lang result for arbitrary commutative algebraic groups.
This is joint work with Jan Dobrowolski and Tingxiang Zou.