Freiman's theorem in an arbitrary nilpotent group

We prove a Freiman–Ruzsa-type theorem valid in an arbitrary nilpotent group. Specifically, we show that a |$K$|-approximate group |$A$| in an |$s$|-step nilpotent group |$G$| is contained in a coset nilprogression of rank at most |$K^{O_s(1)}$| and cardinality at most |$\exp (K^{O_s(1)})|A|$|⁠. To motivate this, we give a direct proof of Breuillard and Green's analogous result for torsion-free nilpotent groups, avoiding the use of Mal’cev's embedding theorem.