Duke Math. J. 170 (18), 3935-3976, (1 December 2021) DOI: 10.1215/00127094-2021-0008
Ya’acov Peterzil, Sergei Starchenko
KEYWORDS: o-minimal, nilmanifolds, Ratner’s theorem, unipotent groups, 03C64, 37A17
Let G be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of , and let Γ be a lattice in G, with the quotient map. For a semialgebraic , and more generally a definable set in an o-minimal structure on the real field, we consider the topological closure of in the compact nilmanifold .
Our theorem describes 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.