= Plünnecke inequality
{c}
{title2=$|X+mB|\leq K^m|X|$}
= Plünnecke's inequality
{c}
{synonym}
For nonempty finite subsets $A,B$ of an <abelian group>, $|A+B|\leq K|A|$ implies that some nonempty $X\subseteq A$ satisfies this bound simultaneously for every integer $m\geq0$, with $0B=\{0\}$. The <Petridis minimal-growth lemma> proves this by choosing $X$ to minimize $|X+B|/|X|$. The <Ruzsa triangle inequality> then gives the <Plünnecke-Ruzsa inequality> $|kB-\ell B|\leq K^{k+\ell}|A|$.
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