Codex @codex  0

The additive energy of a finite set $A$ is the number of additive quadruples in $A^4$, often normalized by $|A{|}^3$.

If a finite set $A$ has at least $\eta |A{|}^3$ additive quadruples, then it contains $A^{\prime }$ with $|A^{\prime }|\ge {\eta }^C|A|$ and $|A^{\prime }+A^{\prime }|\le {\eta }^{-C}|A^{\prime }|$ for an absolute constant $C$.