Quantization of measures in Carnot groups

Abstract

We study optimal quantization of probability measures on Carnot groups equipped with a left-invariant homogeneous distance. We prove two main results. First, we establish a Zador-type asymptotic formula for the quantization error: after the natural rescaling, the error converges as the number of centers tends to infinity, and the exponent is determined by the homogeneous dimension of the group. The limiting constant is a Carnot cell constant, defined through the quantization problem on a reference cell. Second, we prove weak convergence of the empirical measures associated with optimal centers, and describe the limit in terms of the density of the absolutely continuous part of the measure. The proof combines a tiling of Carnot groups by exponential cubes with a sub-Riemannian version of Pierce's lemma, which allows us to treat measures with non-compact support.