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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 KOs(1) and cardinality at most exp(KOs(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.