  We show that an arbitrary nilprogression can be approximated by a proper
coset nilprogression in upper-triangular form. This can be thought of as a
nilpotent version of the Freiman-Bilu result that a generalised arithmetic
progression can be efficiently contained in a proper generalised arithmetic
progression, and indeed an important ingredient in the proof is a Lie-algebra
version of the geometry-of-numbers argument at the centre of that result. We
also 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 $|S^n|\le
Mn^D$ at some sufficiently large scale $n$ then $S$ exhibits polynomial growth
of 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 important
ingredient in a forthcoming companion paper in which we reprove and sharpen a
result about scaling limits of vertex-transitive graphs of polynomial growth
due to Benjamini, Finucane and the first author. We also note that our
arguments imply that every approximate group has a large subset with a large
quotient that is Freiman isomorphic to a subset of a torsion-free nilpotent
group of bounded rank and step.
