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.
Guivarc’h, Croissance polynomiale et p´ eriodes des fonctions harmoniques,Bulletin de la Soci´ et´ e Math´ ematique de France101 (1973), 333–379
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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.