 Breuillard-Green-Tao structure theorem for approximate groups

For every finite $K$-approximate group $A$ in a group $G$, there are subgroups $H⊴C<G$ such that$C/H$ is a nilpotent group of class $O_K(1)$, and $A$ is contained in the union of $O_K(1)$ left cosets of $C$.