Limits of definable families and dilations in nilmanifolds

TL;DR

This paper investigates the limits of definable families in nilmanifolds, establishing conditions under which the Hausdorff limits are either the entire space or smaller subgroups, with special focus on polynomial dilations.

Contribution

It introduces a method to determine Hausdorff limits of definable families in nilmanifolds using finitely many algebraic subgroups, extending understanding of dilations in these spaces.

Findings

Hausdorff limits depend on subgroup conditions

Identifies finitely many algebraic subgroups associated with families

Provides detailed analysis for polynomial dilations

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.