We follow part of the proof of [@bg.sol Corollary 1.5]. It follows from [@bgt.lin Theorem 2.5] that $A$ is contained in a union of at most $K^{O_n(1)}$ left cosets of a nilpotent subgroup $N$ of $GL_n({\mathbb{C}})$ of step at most $n-1$. By \[thm:malcev\] we may assume that $N\subset{\text{\textup{Upp}}}_n({\mathbb{C}})$, and then by \[prop:red.tf\] we may assume that there exists a torsion-free $(n-1)$-step nilpotent group $\Gamma$ such that $N={\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma$. \[lem:fibre.pigeonhole\] then implies that $A$ is contained in a union of at most $K^{O_n(1)}$ left translates of $A^2\cap({\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma)$.

Set $B=A^2\cap\left(\left[-\frac{1}{8},\frac{1}{8}\right]^n\times\Gamma\right)$. The set $\left[-\frac{1}{16},\frac{1}{16}\right]^n\times\Gamma$ is a $2^n$-approximate group; since $(\left[-\frac{1}{16},\frac{1}{16}\right]^n\times\Gamma)^2=\left[-\frac{1}{8},\frac{1}{8}\right]^n\times\Gamma$ and $(\left[-\frac{1}{16},\frac{1}{16}\right]^n\times\Gamma)^8={\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma$, \[lem:slicing\] therefore implies that $B$ is a $2^{3n}K^3$-approximate group, and that $A^2\cap({\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma)$ is contained in a union of at most $2^{7n}K$ left translates of $B$.

Let ${\varphi}:{\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma\to\left(-\frac{1}{2},\frac{1}{2}\right]^n\times\Gamma\subset{\mathbb{R}}^n\times\Gamma$ be the obvious lift. The restriction of ${\varphi}$ to $\left[-\frac{1}{8},\frac{1}{8}\right]^n\times\Gamma$ is a Freiman $3$-homomorphism, so \[lem:fr.hom.ap.grp\] implies that ${\varphi}(B)$ is a $2^{3n}K^3$-approximate subgroup of the torsion-free $s$-step nilpotent group ${\mathbb{R}}^n\times\Gamma$. \[cor:ruzsa,cor:chang.ag\] therefore imply the existence of a nilprogression $Q_1\subset{\varphi}(B)^{e^{O(n^3)}\log^{O(n^2)}2K}$ of rank at most $e^{O(n^2)}\log^{O(n)}2K$ such that ${\varphi}(B)$ is contained in a union of at most $\exp(e^{O(n^2)}\log^{O(n)}2K)$ left translates of $Q_1$, and a nilprogression $Q_2\subset