Appears in collection : 2014 - T1 - WS1 - Asymptotic properties of groups

We consider the orbits of lattice subgroups of semisimple groups acting on homogeneous spaces. We will give general lower and upper bounds on the rate of approximation of a point on the space by a generic orbit. We will then give a sufficient criterion of when these bounds match, and describe many cases in which the criterion holds. This yields best possible results in a host of natural Diophantine approximation problems on homogeneous algebraic varieties. Based on joint Work with A. Ghosh and A. Gorodnik