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Asymptotics of Best-Packing on Rectifiable Sets

S. V. Borodachov·D. P. Hardin·E. B. Saff
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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.

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S. V. Borodachov, D. P. Hardin, E. B. Saff. 2006-05-05. Asymptotics of Best-Packing on Rectifiable Sets. https://arxiv.org/abs/math-ph/0605021

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