Approximate subgroups of residually nilpotent groups

We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is of bounded step. Combined with an earlier result of the author, this implies that A is contained in boundedly many translates of a coset nilprogression of bounded rank and step. The bounds are effective and depend only on K; in particular, if G is nilpotent they do not depend on the step of G. As an application we show that there is some absolute constant c such that if G is a residually nilpotent group, and if there is an integer [Formula: see text] such that the ball of radius n in some Cayley graph of G has cardinality bounded by [Formula: see text] , then G is virtually [Formula: see text] -step nilpotent.