Additive energy produces a large subspace in a fourfold difference set
= 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 quadruple>[additive quadruples], then $2A-2A$ contains a vector subspace of size at least $c'(c,p)|A|$. Combine the <Balog-Szemerédi-Gowers theorem>, the <Freiman-Ruzsa theorem over a finite field>, and the <Finite-field Bogolyubov lemma>.