Limits of definable families and dilations in nilmanifolds

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