 Solution

The hypothesis says that the additive energy satisfies $E(A)\ge c|A{|}^3$. The Balog-Szemerédi-Gowers theorem supplies $A^{\prime }\subseteq A$ withBy the Freiman-Ruzsa theorem over a finite field, $A^{\prime }$ lies in a subspace $H$ withThus $A^{\prime }$ has density at least $C_2^{-1}$ in $H$. Applying the Finite-field Bogolyubov lemma inside $H$ gives a subspaceof codimension bounded in terms of $c,p$ alone. Thereforewhich is the additive energy produces a large subspace in a fourfold difference set result.