O-minimal flows on nilmanifolds

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