Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 129 / 1 / a / Solution

Choose a nonempty $X\subseteq A$ for which $|X+A|/|X|$ is minimal, and write this minimum as $K^{\prime }$. Since $X=A$ is an available choice, $K^{\prime }\le K$. The Petridis minimal-growth lemma givesfor every finite $C$. Taking $C=(t-1)A$ and iterating yieldsfor every integer $t\ge 1$.