Limits of definable families and dilations in nilmanifolds

Abstract

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