= Large lifted product from a coset progression
Let $\pi:G\to G/[G,G]$ be the <abelianization> map and let $A$ be a finite $K$-<approximate group>. Suppose a coset progression $HP(x_1,\ldots,x_r;L_1,\ldots,L_r)$ lies in $\pi(A^4)$ and has size at least $\delta|\pi(A)|$. Then
$$
\left|
\bigl(A^{16}\cap\pi^{-1}(H)\bigr)
\prod_{i=1}^r\bigl(A^{22}\cap\pi^{-1}(\langle x_i\rangle)\bigr)
\right|\geq\delta|A|.
$$
A section of $\pi$ is multiplicative up to a bounded power of $A$ inside the <commutator subgroup>; successively separating the subgroup and progression coordinates proves the inclusion needed for this estimate.
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