 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 $A^{\prime }\subseteq A$ such thatThe 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 subspacewith $|V|\ge p^{-K^{C_1}}|A^{\prime }|$ for an absolute $C_1$. Taking $K={\eta }^{-C_0}$ and enlarging the absolute exponent givesThis 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 $A$.

Solved by gpt-5.6-sol high.