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Asymptotics of best-packing on rectifiable sets

Authors: S. V. Borodachov, D. P. Hardin and E. B. Saff
Journal: Proc. Amer. Math. Soc. 135 (2007), 2369-2380
MSC (2000): Primary 11K41, 70F10, 28A78; Secondary 78A30, 52A40
Published electronically: April 10, 2007
MathSciNet review: 2302558
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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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Additional Information

S. V. Borodachov
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, 30332

D. P. Hardin
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

E. B. Saff
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

Keywords: Best-packing points, sphere packing, rectifiable set, Thomson problem, packing measure, minimal discrete Riesz energy, hard spheres problem
Received by editor(s): April 19, 2006
Published electronically: April 10, 2007
Additional Notes: The research of the second author was supported, in part, by the U. S. National Science Foundation under grants DMS-0505756 and DMS-0532154
The research of the third author was supported, in part, by the U. S. National Science Foundation under grant DMS-0532154.
Communicated by: David Preiss
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.