Few products, many h-fold sums

Abstract

Improving upon a technique of Croot and Hart, we show that for every $h$, there exists an $\epsilon >0$ such that if $A\subseteq \mathbb{R}$ is sufficiently large and $|A.A|\le |A{|}^{1+\epsilon }$, then $|hA|\ge |A{|}^{\Omega (e^{\sqrt{c\log h}})}$.

19 pages