Few products, many h-fold sums

TL;DR

This paper improves bounds on the size of h-fold sumsets for sets with small product sets, showing that such sumsets grow rapidly with the set size, extending previous work by Croot and Hart.

Contribution

It establishes a new exponential lower bound on the size of h-fold sumsets for sets with small multiplicative doubling, generalizing prior results.

Findings

For large sets with small product sets, h-fold sumsets are significantly large.

The lower bound on sumset size grows exponentially with ",

The result applies for all h, with bounds depending on ",

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 $|hA| \ge |A|^{\Omega(e^{\sqrt{c\log{h}}})}$.