Homogeneous orbit closures and applications

We give new classes of examples of orbits of the diagonal group in the space of unit volume lattices in R^d for d > 2 with nice (homogeneous) orbit closures, as well as examples of orbits with explicitly computable but irregular orbit closures. We give Diophantine applications to the former, for instance we show that if x is the cubic root of 2 then for any y,z in R liminf |n|<nx-y><nx^2-z>=0 (as |n| goes to infinity), where <c> denotes the distance of a real number c to the integers.