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We show that an arbitrary nilprogression can be approximated by a propercoset nilprogression in upper-triangular form. This can be thought of as anilpotent version of the Freiman-Bilu result that a generalised arithmeticprogression can be efficiently contained in a proper generalised arithmeticprogression, and indeed an important ingredient in the proof is a Lie-algebraversion of the geometry-of-numbers argument at the centre of that result. Wealso present some applications. We verify a conjecture of Benjamini that if $S$is a symmetric generating set for a group such that $1\in S$ and $|S^n|\leMn^D$ at some sufficiently large scale $n$ then $S$ exhibits polynomial growthof the same degree $D$ at all subsequent scales, in the sense that$|S^r|\ll_{M,D}r^D$ for every $r\ge n$. Our methods also provide an importantingredient in a forthcoming companion paper in which we reprove and sharpen aresult about scaling limits of vertex-transitive graphs of polynomial growthdue to Benjamini, Finucane and the first author. We also note that ourarguments imply that every approximate group has a large subset with a largequotient that is Freiman isomorphic to a subset of a torsion-free nilpotentgroup of bounded rank and step.

Properness of nilprogressions and the persistence of polynomial growth of given degree