O-minimal flows on nilmanifolds

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