Additive energy

Summary

The additive energy of two finite sets $A$ and $B$ in an abelian group counts the quadruples $(a_1, a_2, b_1, b_2)$ with $a_1 + b_1 = a_2 + b_2$. It measures how often sums repeat, so it is large when the sumset $A + B$ is small. It is a more robust measure of additive structure than the size of the sumset. The Balog–Szemerédi–Gowers theorem turns large energy into a small sumset for large subsets.