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Discrepancy of Minimal Riesz Energy Points

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Abstract

We find upper bounds for the spherical cap discrepancy of the set of minimizers of the Riesz s-energy on the sphere \(\mathbb S^d.\) Our results are based on bounds for a Sobolev discrepancy introduced by Thomas Wolff in an unpublished manuscript where estimates for the spherical cap discrepancy of the logarithmic energy minimizers in \(\mathbb S^2\) were obtained. Our result improves previously known bounds for \(0\le s<2\) and \(s\ne 1\) in \(\mathbb S^2,\) where \(s=0\) is Wolff’s result, and for \(d-t_0<s<d\) with \(t_0\approx 2.5\) when \(d\ge 3\) and \(s\ne d-1.\)

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Authors and Affiliations

  1. Departament de Matemàtiques i Informàtica, Barcelona Graduate School of Mathematics (BGSMath), Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    Jordi Marzo

  2. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Campus Diagonal Besòs, Edifici A (EEBE), Av. Eduard Maristany 16, 08019, Barcelona, Spain

    Albert Mas

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  1. Jordi Marzo
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  2. Albert Mas
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Correspondence to Jordi Marzo.

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Communicated by Arno Kuijlaars.

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The authors are grateful to Joaquim Ortega-Cerdà and Carlos Beltrán for enlightening discussions on the subject matter of this paper, and to Alexandre Eremenko for information about Wolff’s manuscript.

The first author has been supported by Grant MTM2017-83499-P by the Ministerio de Economía y Competitividad, Gobierno de España and by the Generalitat de Catalunya (Project 2017 SGR 358). The second author was partially supported by MTM2017-84214 and MTM2017-83499 projects of the MCINN (Spain), 2017-SGR-358 project of the AGAUR (Catalunya), and ERC-2014-ADG project HADE Id. 669689 (European Research Council).

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Marzo, J., Mas, A. Discrepancy of Minimal Riesz Energy Points. Constr Approx 54, 473–506 (2021). https://doi.org/10.1007/s00365-021-09534-5

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  • DOI: https://doi.org/10.1007/s00365-021-09534-5

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