= Polynomial progression in a high-energy fourfold difference set
{title2=$|P|\geq|A|^{\gamma(\theta)}$}
For every $\theta>0$, some $\gamma(\theta)>0$ has this property: any nonempty finite $A\subseteq\mathbb Z$ with $E(A)\geq\theta|A|^3$ has an <arithmetic progression> of length at least $|A|^{\gamma(\theta)}$ in $2A-2A$. The <small-difference-set form of the Balog-Szemerédi-Gowers theorem>, <Ruzsa modelling lemma>, <cyclic Bogolyubov lemma> and <nonwrapping progression in a cyclic Bohr set> prove it. An order-eight <Freiman s-isomorphism> suffices to lift the progression because its consecutive second-difference equations expand into equalities of eight-term sums.
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