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Transactions of the American Mathematical Society

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Geodesic distance Riesz energy on the sphere


Authors: Dmitriy Bilyk and Feng Dai
Journal: Trans. Amer. Math. Soc. 372 (2019), 3141-3166
MSC (2010): Primary 11K38, 74G65; Secondary 42C10, 33C55
DOI: https://doi.org/10.1090/tran/7711
Published electronically: November 26, 2018
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Abstract: We study energy integrals and discrete energies on the sphere, in particular, analogues of the Riesz energy with the geodesic distance in place of the Euclidean, and we determine that the range of exponents for which uniform distribution optimizes such energies is different from the classical case. We also obtain a very general form of the Stolarsky principle, which relates discrete energies to certain $ L^2$ discrepancies, and prove optimal asymptotic estimates for both objects. This leads to sharp asymptotics of the difference between optimal discrete and continuous energies in the geodesic case, as well as new proofs of discrepancy estimates.


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Additional Information

Dmitriy Bilyk
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55408
Email: dbilyk@math.umn.edu

Feng Dai
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: fdai@ualberta.ca

DOI: https://doi.org/10.1090/tran/7711
Received by editor(s): November 9, 2017
Received by editor(s) in revised form: May 20, 2018, and July 1, 2018
Published electronically: November 26, 2018
Additional Notes: The stay of the first author at CRM (Barcelona) has been sponsored by NSF grant DMS 1613790. The work of the first author is partially supported by the Simons Foundation collaboration grant and NSF grant DMS 1665007.
The work of the second author is partially supported by NSERC Canada under grant RGPIN 04702
Article copyright: © Copyright 2018 American Mathematical Society