  The present paper establishes upper and lower bounds on the speed of
approximation in a wide range of natural Diophantine approximation problems.
The upper and lower bounds coincide in many cases, giving rise to optimal
results in Diophantine approximation which were inaccessible previously. Our
approach proceeds by establishing, more generally, upper and lower bounds for
the rate of distribution of dense orbits of a lattice subgroup $\Gamma $ in a
connected Lie (or algebraic) group $G$, acting on suitable homogeneous spaces
$G/H$. The upper bound is derived using a quantitative duality principle for
homogeneous spaces, reducing it to a rate of convergence in the mean ergodic
theorem for a family of averaging operators supported on $H$ and acting on
$G/\Gamma$. In particular, the quality of the upper bound on the rate of
distribution we obtain is determined explicitly by the spectrum of $H$ in the
automorphic representation on $L^2(\Gamma\setminus G)$. We show that the rate
is best possible when the representation in question is tempered, and show that
the latter condition holds in a wide range of examples.
