Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets

Authors:
S. V. Borodachov, D. P. Hardin and E. B. Saff

Journal:
Trans. Amer. Math. Soc. **360** (2008), 1559-1580

MSC (2000):
Primary 11K41, 70F10, 28A78; Secondary 78A30, 52A40

DOI:
https://doi.org/10.1090/S0002-9947-07-04416-9

Published electronically:
October 17, 2007

MathSciNet review:
2357705

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Abstract: Given a closed -rectifiable set embedded in Euclidean space, we investigate minimal weighted Riesz energy points on ; that is, points constrained to and interacting via the weighted power law potential , where is a fixed parameter and is an admissible weight. (In the unweighted case () such points for fixed tend to the solution of the best-packing problem on as the parameter .) Our main results concern the asymptotic behavior as of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution with respect to -dimensional Hausdorff measure on , our results provide a method for generating -point configurations on that are ``well-separated'' and have asymptotic distribution as .

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Additional Information

**S. V. Borodachov**

Affiliation:
Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240

Address at time of publication:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

Email:
sergiy.v.borodachov@vanderbilt.edu, borodasv@math.gatech.edu

**D. P. Hardin**

Affiliation:
Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240

Email:
doug.hardin@vanderbilt.edu

**E. B. Saff**

Affiliation:
Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240

Email:
edward.b.saff@vanderbilt.edu

DOI:
https://doi.org/10.1090/S0002-9947-07-04416-9

Keywords:
Minimal discrete Riesz energy,
best-packing,
Hausdorff measure,
rectifiable sets,
non-uniform distribution of points,
power law potential,
separation radius

Received by editor(s):
February 10, 2006

Published electronically:
October 17, 2007

Additional Notes:
The research of the first author was conducted as a graduate student under the supervision of E.B. Saff and D. P. Hardin at Vanderbilt University.

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.

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.