Asymptotics of Best-packing on Rectifiable Sets

We investigate the asymptotic behavior, as N grows, of the largest minimal pairwise distance of N points restricted to an arbitrary compact rectifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we compare best-packing configurations with minimal Riesz s-energy configurations and determine the s-th root asymptotic behavior (as s→∞) of the minimal energy constants. We show that the upper and the lower dimension of a set defined through the Riesz energy or best-packing coincides with the upper and lower Minkowski dimension, respectively. For certain sets in R of integer Hausdorff dimension, we show that the limiting behavior of the best-packing distance as well as the minimal s-energy for large s is different for different subsequences of the cardinalities of the configurations.