A translation principle turns products of finite subsets of the Baumslag-Solitar group BS(1,n) into Minkowski sums of dilates of integer sets, so that |S^2| for S contained in a coset of the cyclic subgroup equals |n*A + A|. Using this correspondence, a small doubling hypothesis on a finite non-abelian subset S of the monoid BS+(1,2) forces S to lie in a single coset and to be contained in a short geometric progression, the group-theoretic analogue of Freiman-style arithmetic-progression conclusions. The student learns how doubling constants constrain algebraic structure and how extended inverse problems relax exact extremal bounds while still yielding structural conclusions.
This paper by Freiman, Herzog, Longobardi, Maj, and Stanchescu addresses inverse problems in additive combinatorics: given structural information about a sumset A+A (or more generally, Minkowski sums…