Large sumsets from medium-sized subsets

The classical Cauchy–Davenport inequality gives a lower bound for the size of the sum of two subsets of ${\mathbb{Z}}_p$, where $p$ is a prime. Our main aim in this paper is to prove a considerable strengthening of this inequality, where we take only a small number of points from each of the two subsets when forming the sum. One of our results is that there is an absolute constant $c>0$ such that if $A$ and $B$ are subsets of ${\mathbb{Z}}_p$ with $\mathrm{\mid }A\mathrm{\mid }=\mathrm{\mid }B\mathrm{\mid }=n\le p\mathrm{/}3$ then there are subsets $A^{\mathrm{\prime }}\subset A$ and $B^{\mathrm{\prime }}\subset B$ with $\mathrm{\mid }{A}^{\mathrm{\prime }}\mathrm{\mid }=\mathrm{\mid }{B}^{\mathrm{\prime }}\mathrm{\mid }\le c\sqrt{n}$ such that $\mathrm{\mid }{A}^{\mathrm{\prime }}+B^{\mathrm{\prime }}\mathrm{\mid }\ge 2n-1$. In fact, we show that one may take any sizes one likes: as long as $c_1$ and $c_2$ satisfy $c_1{c}_2\ge cn$ then we may choose $\mathrm{\mid }{A}^{\mathrm{\prime }}\mathrm{\mid }=c_1$ and $\mathrm{\mid }{B}^{\mathrm{\prime }}\mathrm{\mid }=c_2$. We prove related results for general abelian groups.