 Plünnecke-Ruzsa inequality

Let $A,B$ be nonempty finite subsets of an abelian group, and suppose $|A+B|\le K|A|$. For nonnegative integers $\ell ,m$,where $0B=\{0\}$. A Plünnecke inequality minimizing subset $X\subseteq A$ has $|X+jB|\le K^j|X|$ for every nonnegative $j$. The Ruzsa triangle inequality therefore givesThe empty-set exclusion is needed for the positive denominator and the growth hypothesis.^[ref]