Breuillard-Green-Tao structure theorem for approximate groups

For every finite $K$-approximate group $A$ in a group $G$, there are subgroups $H⊴C\le G$ such that $H\subseteq A^4$, the quotient group $C/H$ is nilpotent of class $O_K(1)$, and $A$ is contained in the union of $O_K(1)$ left cosets of $C$.

The structural subgroup $C$ may equal $G$. For example, if $A=G$ is a cyclic group of large prime order, then $A$ is a $1$-approximate group. Its only proper subgroup is trivial, so a bounded number of cosets of a proper subgroup cannot cover $A$ uniformly in the prime. The full structure theorem provides finer nilpotent progression control; the statement here records its quotient-and-covering consequence.