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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-07-08975-7
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 $\textrm {\textbf {R}}^d$ 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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Additional Information

S. V. Borodachov
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, 30332
MR Author ID: 656604
Email: borodasv@math.gatech.edu

D. P. Hardin
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
MR Author ID: 81245
ORCID: 0000-0003-0867-2146
Email: doug.hardin@vanderbilt.edu

E. B. Saff
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
MR Author ID: 152845
Email: Edward.B.Saff@Vanderbilt.edu

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.