O-minimal flows on nilmanifolds

Let $G$ be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let $\Gamma$ be a lattice in $G$, with $\pi:G\to G/\Gamma$ the quotient map. For a semi-algebraic $X\subseteq G$, and more generally a definable set in an o-minimal structure on the real field, we consider the topological closure of $\pi(X)$ in the compact nilmanifold $G/\Gamma$. Our theorem describes $\mathrm{cl}(\pi(X))$ in terms of finitely many families of cosets of real algebraic subgroups of $G$. The underlying families are extracted from $X$, independently of $\Gamma$. We also prove an equidistribution result in the case of curves.