 Additive energy produces a large subspace in a fourfold difference set

For fixed $c>0$ and prime $p$, if $A\subseteq {\mathbb{F}}_p^N$ has at least $c|A{|}^3$ additive quadruples, then $2A-2A$ contains a vector subspace of size at least $c^{\prime }(c,p)|A|$. Combine the Balog-Szemerédi-Gowers theorem, the Freiman-Ruzsa theorem over a finite field, and the Finite-field Bogolyubov lemma.