If $A+A$ is small then $AAA$ is superquadratic

TL;DR

This paper proves that if a finite set of real numbers has a small sumset, then its triple product set must be significantly larger, establishing a superquadratic growth relationship.

Contribution

It establishes a quantitative link between small sumsets and superquadratic growth of triple product sets for finite real sets.

Findings

Sets with small sumsets have superquadratic triple product sets.

Existence of positive constants c_1 and c_2 linking sumset size and triple product size.

Triple product set size grows faster than quadratic for sets with small sumsets.

Abstract

This note proves that there exists positive constants $c_1$ and $c_2$ such that for all finite $A \subset \mathbb R$ with $|A+A| \leq |A|^{1+c_1}$ we have $|AAA| \gg |A|^{2+c_2}$.