The hypothesis says that the <additive energy> satisfies $E(A)\geq c|A|^3$. The <Balog-Szemerédi-Gowers theorem> supplies $A'\subseteq A$ with
$$
|A'|\geq c_1(c)|A|,
\qquad
|A'+A'|\leq C_1(c)|A'|.
$$
By the <Freiman-Ruzsa theorem over a finite field>, $A'$ lies in a subspace $H$ with
$$
|H|\leq C_2(c,p)|A'|.
$$
Thus $A'$ has density at least $C_2^{-1}$ in $H$. Applying the <Finite-field Bogolyubov lemma> inside $H$ gives a subspace
$$
V\subseteq2A'-2A'\subseteq2A-2A
$$
of codimension bounded in terms of $c,p$ alone. Therefore
$$
|V|\geq c'(c,p)|H|\geq c'(c,p)|A|,
$$
which is the <additive energy produces a large subspace in a fourfold difference set> result.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 129 / 3 / ii / Solution

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>.

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