Proceedings of the American Mathematical Society

Author(s): S. V. Borodachov; D. P. Hardin; E. B. Saff
Journal: Proc. Amer. Math. Soc. 135 (2007), 2369-2380.
MSC (2000): Primary 11K41, 70F10, 28A78; Secondary 78A30, 52A40
Posted: April 10, 2007
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Abstract: We investigate the asymptotic behavior, as

grows, of the largest minimal pairwise distance of

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

-energy configurations and determine the

-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

-energy for large

is different for different subsequences of the cardinalities of the configurations.

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Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11K41, 70F10, 28A78, 78A30, 52A40

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

D. P. Hardin
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: doug.hardin@vanderbilt.edu

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

DOI: 10.1090/S0002-9939-07-08975-7
PII: S 0002-9939(07)08975-7
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
Posted: 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
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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