There is a theorem In Terence Tao's Additive Combinatorics:
Proposition 2.2 (Exact inverse sum set theorem) Suppose that $A$, $B$ are additive sets with common ambient group Z . Then the following are equivalent:
- $|A + B| = |A|$;
- there exists a finite subgroup $G$ of $Z$ such that $B$ is contained in a coset of $G$, and A is a union of cosets of $G$.
It is straight forward to show that 1 implies 2. But how can we get the reverse direction work?