Approximate subgroups of residually nilpotent groups.

We show that a -approximate subgroup of a residually nilpotent group 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 is contained in boundedly many translates of a coset nilprogression of bounded rank and step. The bounds are effective and depend only on ; in particular, if is nilpotent they do not depend on the step of . As an application we show that there is some absolute constant such that if is a residually nilpotent group, and if there is an integer such that the ball of radius in some Cayley graph of has cardinality bounded by , then is virtually -step nilpotent.