In virtually nilpotent groups, finite sets whose powers contain symmetric word balls of radius comparable to h are asymptotic approximate groups, along with a nonabelian semilinear-set version for some infinite sets.
Breuillard, Geometry of locally compact groups of polynomial growth and shape of large balls,Groups, Geometry, and Dynamics8 (2014), 669–732
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On asymptotic approximate groups in nilpotent groups
In virtually nilpotent groups, finite sets whose powers contain symmetric word balls of radius comparable to h are asymptotic approximate groups, along with a nonabelian semilinear-set version for some infinite sets.