Large sumsets from medium-sized subsets

TL;DR

This paper strengthens the classical Cauchy--Davenport inequality by showing that small, medium-sized subsets of cyclic groups can produce large sumsets, with bounds depending on subset sizes.

Contribution

It provides a new bound demonstrating that small subsets can generate large sumsets, extending classical results to more general abelian groups.

Findings

Existence of small subsets with large sumsets in cyclic groups

Generalization of results to arbitrary abelian groups

Quantitative bounds relating subset sizes to sumset size

Abstract

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 $|A|=|B|=n\le p/3$ then there are subsets $A'\subset A$ and $B'\subset B$ with $|A'|=|B'|\le c \sqrt{n}$ such that $|A'+B'|\ge 2n-1$. In fact, we show that one may take any sizes one likes: as long as $c_1$ and $c_2$ satisfy $c_1c_2 \ge cn$ then we may choose $|A'|=c_1$ and $|B'|=c_2$. We prove related results for general abelian groups.