Let G be a Lie group, L a lattice in G, and H a closed subgroup of G.
Suppose that L acts on the homogeneous space G/H with dense orbits.
We would like to measure how dense these orbits actually are, or equivalently, gauge the efficiency of approximation of a general point on G/H by a lattice orbit.
Departing from traditional classical Diophantine approximation, we will
Assume G to be a non-amenable group, for example the group of isometries of hyperbolic space, or the general linear or affine group.
We will present a solution to this problem for lattice actions
on a large class of homogeneous spaces, emphasizing a sufficient condition for when an optimal result holds, and give some examples. The methods involve dynamical arguments, and spectral methods applied to the automorphic representation.
We will then briefly describe the extensive scope of this set-up, and explain some more refined problems related to equidistribution and discrepancy of lattice orbits, as time permits.

Based partly on joint work with Alex Gorodnik and Anish Ghosh, and partly on recent joint work with Mikolaj Fraczyk and Alex Gorodnik.