Asymptotics of Best-Packing on Rectifiable Sets

TL;DR

This paper studies the asymptotic behavior of optimal point configurations on rectifiable sets, linking best-packing and Riesz energy, and reveals dimension-related properties and subsequence-dependent behaviors.

Contribution

It establishes the asymptotic relationship between best-packing and Riesz energy configurations on rectifiable sets, connecting these to Minkowski dimension and subsequence behaviors.

Findings

Best-packing configurations' asymptotics match Minkowski dimension.

Minimal Riesz s-energy constants have s-th root asymptotics as s→∞.

Different subsequences show varied limiting behaviors for sets with integer Hausdorff dimension.

Abstract

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 $s\to \infty)$ 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 ${\rm {\bf R}}^d$ 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.