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

Abstract We prove a number field analogue of a result of C. McMullen that the set of badly approximable numbers is absolute winning. We also prove a weighted version. We use this to prove an instance of a conjecture of An, Guan and Kleinbock [4] . Namely, let G : = SL 2 ( R ) × … × SL 2 ( R ) and Γ be a lattice in G. We show that the set of points on G / Γ whose orbits under a one parameter Ad-semisimple subgroup of G are bounded, form a hyperplane absolute winning set.