Plünnecke-Ruzsa inequality

If finite sets satisfy $|A+B|\le K|A|$, then the Plünnecke-Ruzsa inequality bounds iterated sumsets and difference sets by

If a nonempty finite set $X$ minimizes $|X+B|/|X|$ among the nonempty subsets of $A$, with minimum $K^{\prime }$, thenfor every finite set $C$. Iteration is a short proof of the Plünnecke-Ruzsa inequality.

For finite subsets $A,B,C$ of an abelian group with $A$ nonempty,An injective encoding chooses one representation of each element of $B-C$ and translates it by every element of $A$.