Arithmetic Homogeneous Spaces

We will describe some recent ergodic theorems for general families of averages on semisimple Lie groups, and explain how they can be used to 1) Solve the lattice point counting problem for general domains in the group, with explicit estimate of the error term, 2) Prove pointwise ergodic theorems for arbitrary measure preserving actions of the lattice, with explicit rate of convergence to the ergodic mean when the action has a spectral gap, 3) Prove equidistribution and uniform convergence to the ergodic mean in general ergodic isometric actions of the lattice.