Raghunathan conjectured that If G is a Lie group, Gamma a lattice, p in G/Gamma, and U an (ad-)unipotent group then the closure of U.p is homogeneous (a periodic orbit of a subgroup of G). This conjecture was proved by Ratner in the early 90's via the classification of invariant measures; significant special cases were proved earlier by Dani and Margulis using a different, topological dynamics approach. Neither of these proofs is effective, nor do they provide rates --- e.g. if p is generic in the sense that it does not lie on a
periodic orbit of any proper subgroup L<G with U<=L, an estimate (possibly depending on diophantine-type properties of the pair (p,U))) how large a piece of an orbit is needed so that it comes within distance epsilon of any point in a given compact subset of G/Gamma