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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Asymptotics of best-packing on rectifiable sets
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by S. V. Borodachov, D. P. Hardin and E. B. Saff PDF
Proc. Amer. Math. Soc. 135 (2007), 2369-2380 Request permission

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
  • 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
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2302558