  We prove mean and pointwise ergodic theorems for the action of a discrete
lattice subgroup in a connected algebraic Lie group, on infinite volume
homogeneous algebraic varieties. Under suitable necessary conditions, our
results are quantitative, namely we establish rates of convergence in the mean
and pointwise ergodic theorems, which can be estimated explicitly. Our results
give a precise and in most cases optimal quantitative form to the duality
principle governing dynamics on homogeneous spaces. We illustrate their scope
in a variety of equidistribution problems.
