One major area of study in additive combinatorics are inverse problems: given the size of the sumset A + B is small, what can we say about the structures of A and B?
In the case of the integers, the classical Freiman's theorem provides a partial answer to this question in terms of multi-dimensional arithmetic progressions.
Now we have the inequality for the cardinality of the sum set A + B, it is natural to ask the inverse problem, namely under what conditions on A and B does the equality hold?
For instance using Plünnecke–Ruzsa inequality, we are able to prove a version of Freiman's Theorem in finite fields.
Let A and B be finite subsets of an abelian group; then the sum set is defined to be For example, we can write {1,2,3,4} + {1,2,3} = {2,3,4,5,6,7}.