Asymptotics of best-packing on rectifiable sets

Borodachov, S.; Hardin, D.; Saff, E.

doi:10.1090/S0002-9939-07-08975-7

Asymptotics of best-packing on rectifiable sets
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by S. V. Borodachov, D. P. Hardin and E. B. Saff

Proc. Amer. Math. Soc. 135 (2007), 2369-2380

DOI: https://doi.org/10.1090/S0002-9939-07-08975-7

Published electronically: April 10, 2007

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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 $\textrm {\textbf {R}}^d$ 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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