Small subsets with large sumset: Beyond the Cauchy--Davenport bound

TL;DR

This paper investigates how small subsets of an abelian group can produce large sumsets, surpassing classical bounds, and resolves conjectures related to sumset sizes in cyclic groups and high-dimensional settings.

Contribution

It introduces bounds on sumset sizes generated by small subsets, proving new results that extend beyond classical theorems and resolving existing conjectures.

Findings

A small subset of size at most s can produce a sumset of size  \, ext{min}(\, ext{large doubling} \, ext{and} \, s) \, |A|.

Three elements from B suffice to guarantee a sumset nearly as large as the sum of sizes of A and B.

For sets with bounded doubling, a small subset can generate the entire sumset A+A.

Abstract

For a subset $A$ of an abelian group $G$, given its size $|A|$, its doubling $\kappa=|A+A|/|A|$, and a parameter $s$ which is small compared to $|A|$, we study the size of the largest sumset $A+A'$ that can be guaranteed for a subset $A'$ of $A$ of size at most $s$. We show that a subset $A'\subseteq A$ of size at most $s$ can be found so that $|A+A'| = \Omega(\min(\kappa^{1/3},s)|A|)$. Thus a sumset significantly larger than the Cauchy--Davenport bound can be guaranteed by a bounded size subset assuming that the doubling $\kappa$ is large. Building up on the same ideas, we resolve a conjecture of Bollob\'as, Leader and Tiba that for subsets $A,B$ of $\mathbb{Z}_p$ of size at most $\alpha p$ for an appropriate constant $\alpha>0$, one only needs three elements $b_1,b_2,b_3\in B$ to guarantee $|A+\{b_1,b_2,b_3\}|\ge |A|+|B|-1$. Allowing the use of larger subsets $A'$, we show that for sets $A$ of bounded doubling, one only needs a subset $A'$ with $o(|A|)$ elements to guarantee that $A+A'=A+A$. We also address another conjecture and a question raised by Bollob\'as, Leader and Tiba on high-dimensional analogs and sets whose sumset cannot be saturated by a bounded size subset.