The Additive Energy of a Set Is at Least $|A|^4/|G|$

For a finite abelian group $G$ and $A G$, the additive energy satisfies $E(A) |A|^4/|G|$. The proof is Cauchy–Schwarz applied to the representation counts, which sum to $|A|^2$ across $|G|$ points: squares cannot sum to less than the square of the mean. Two things are worth pinning alongside it. The bound is often met in the weaker form $E(A) ^3|G|$ with $ = |A|/|G|$; that form follows from this one and falls short of it by a factor of exactly $|A||G|$. And the bound is sometimes attached to the name Bloom–Sisask, which is a different theorem about a different object –- it bounds a progression-free subset of $[N]$, and no progression occurs anywhere below.