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 $A$.

For nonempty finite subsets $A,B,C$ of an arbitrary group,Choose one representation $x=b_x{c}_x^{-1}$ for each $x\in B{C}^{-1}$. The map $(a,x)\mapsto (a{b}_x^{-1},a{c}_x^{-1})$ is injective.

If a finite subset $A$ of a group satisfies $|A^3|\le K|A|$, then repeated use of the Noncommutative Ruzsa triangle inequality gives

Let $H$ be a finite subgroup and choose $x$ with $H\cap xH{x}^{-1}=\{1\}$. For $A=H\cup \{x\}$, the set $A^2$ has size at most $3|H|+1$, while $A^3$ contains the double coset $HxH$ of size $|H{|}^2$. Thus bounded doubling alone gives no tripling bound in arbitrary groups.