= Solution
The hypothesis says that the normalized <additive energy> of $A$ is at least $\eta$. By the <Balog-Szemerédi-Gowers theorem>, for an absolute $C_0$ there is $A'\subseteq A$ such that
$$
|A'|\geq\eta^{C_0}|A|,
\qquad
|A'+A'|\leq\eta^{-C_0}|A'|.
$$
The finite-field Bogolyubov-Ruzsa consequence of the <Freiman-Ruzsa theorem over a finite field> says that a set of doubling at most $K$ has a <vector subspace>
$$
V\subseteq A'+A'-A'-A'
$$
with $|V|\geq p^{-K^{C_1}}|A'|$ for an absolute $C_1$. Taking $K=\eta^{-C_0}$ and enlarging the absolute exponent gives
$$
V\subseteq A+A-A-A,
\qquad
|V|\geq p^{-\eta^{-C}}|A|.
$$
This is an energy form of the <Bogolyubov lemma>: the usual lemma assumes positive density in an ambient group, whereas the <Balog-Szemerédi-Gowers theorem> first extracts a dense structured model from the many additive quadruples. The resulting bound depends on the energy parameter rather than on the possibly tiny ambient density of $A$.
Back to article page
