  The quantization problem looks for best approximations of a probability
measure on a given metric space by finitely many points, where the
approximation error is measured with respect to the Wasserstein distance. On
particular smooth domains, such as $\mathbb{R}^d$ or complete Riemannian
manifolds, the quantization error is known to decay polynomially as the number
of points is taken to infinity, provided the measure satisfies an integral
condition which controls the amount of mass outside compact sets. On Riemannian
manifolds, the existing integral condition involves a quantity measuring the
growth of the exponential map, for which the only available estimates are in
terms of lower bounds on sectional curvature.
  In this paper, we provide a more general integral condition for the
asymptotics of the quantization error on Riemannian manifolds, given in terms
of the growth of the covering numbers of spheres, which is purely metric in
nature and concerns only the large-scale growth of the manifold. We further
estimate the covering growth of manifolds in two particular cases, namely lower
bounds on the Ricci curvature and geometric group actions by a discrete group
of isometries. These estimates can themselves generalize beyond manifolds, and
hint at a future treatment of asymptotic quantization also on non-smooth metric
measure spaces.
