Asymptotics of best-packing on rectifiable sets

Abstract

We investigate the asymptotic behavior, as N grows, of the largest minimal pairwise distance of N points restricted to an arbitrary compact rec-tifiable 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 d 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. 1. Preliminaries The problem of finding a configuration of N points on the sphere with the minimal pairwise distance between the points being as large as possible is classical and is known as Tammes's problem or the hard spheres problem. When formulated for the whole Euclidean space, the analogous problem is that of finding a collection (or packing) of non-overlapping equal balls with the largest density. More information on this problem and its generalizations can be found in [2], [4], [7], [19]. In the present paper we investigate the best-packing problem on certain classes of "non-smooth" sets. 1.1. Best-packing problem. We denote by R d the embedding space, reserving the symbol d for the dimension of the set being considered. For a collection of N distinct points ω N = {y 1 ,. .. , y N } ⊂ R d we set δ(ω N) := min 1≤i =j≤N |y i − y j |,