Limits of definable families and dilations in nilmanifolds

Let G be a unipotent group and F=\Fₜ: t (0, ) \ a family of subsets of G, with F definable in an o-minimal expansion of the real field. Given a lattice G, we study the possible Hausdorff limits of (F) in G/ as t tends to (here: G G/ is the canonical projection). Towards a solution, we associate to F finitely many real algebraic subgroups L G, and, uniformly in, determine if the only Hausdorff limit at is G/, depending on whether L^=G or not. The special case of polynomial dilations of a definable set is treated in details.