The problem of quantization of measures looks for best approximations of probability measures on a metric space by discrete measures supported on N points, where the error of approximation is measured with respect to the Wasserstein distance. Zador’s theorem states that, for measures on \({\mathbb {R}}^d\) or d-dimensional Riemannian manifolds satisfying appropriate integrability conditions, the quantization error decays to zero as \(N \rightarrow \infty \) at the rate \(N^{-1/d}\) . In this paper, we provide a general treatment of the asymptotics of quantization on metric measure spaces \((X, \nu )\) . We show that a weaker version of Zador’s theorem involving the Hausdorff densities of \(\nu \) holds also in this general setting. We also prove Zador’s theorem in full for appropriate m-rectifiable measures on Euclidean space, answering a conjecture by Graf and Luschgy in the affirmative. For both results, the higher integrability conditions of Zador’s theorem are replaced with a general notion of (p, s)-quantizability, which follows from Pierce-type (non-asymptotic) upper bounds on the quantization error, and we also prove multiple such bounds at the level of metric measure spaces.
