= Breuillard-Green-Tao structure theorem for approximate groups
{c}
For every finite $K$-<approximate group> $A$ in a <group> $G$, there are <subgroups> $H\trianglelefteq C\leq G$ 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.
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