Bounded orbits of diagonalizable flows on finite volume quotients of products of SL2(R)

Abstract

We prove a number field analogue of W. M. Schmidt's conjecture on the intersection of weighted badly approximable vectors and use this to prove an instance of a conjecture of An, Guan and Kleinbock. Namely, let G := SL_2(\mathbb{R}) \times \dots \times SL_2(\mathbb{R}) and \Gamma be a lattice in G. We show that the set of points on G/\Gamma whose forward orbits under a one parameter Ad-semisimple subsemigroup of G are bounded, form a hyperplane absolute winning set.