Few products, many h-fold sums

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