On orbits of unipotent flows on homogeneous spaces, II

Abstract

We show that if (ut) is a one-parameter subgroup of SL (n, [formula omitted]) consisting of unipotent matrices, then for any ε > 0 there exists a compact subset K of SL(n, [formula omitted])/SL(n, ℤ) such that the following holds: for any g [formula omitted] SL(n, [formula omitted]) either m({t [formula omitted] [0, T] | utg SL (n, ℤ) [formula omitted] K}) > (1 – ε)T for all large T (m being the Lebesgue measure) or there exists a non-trivial (g−1utg)-invariant subspace defined by rational equations. Similar results are deduced for orbits of unipotent flows on other homogeneous spaces. We also conclude that if G is a connected semisimple Lie group and Γ is a lattice in G then there exists a compact subset D of G such that for any closed connected unipotent subgroup U, which is not contained in any proper closed subgroup of G, we have G = DΓ U. The decomposition is applied to get results on Diophantine approximation. © 1986, Cambridge University Press. All rights reserved.