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≥3 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, for all γ,δ∈R iminf _|n|→∞|n|√ ³2-γn ³4- δ=0, where denotes the distance of a real number c to the integers.