Abstract
In Beltrán and Etayo (J. Complexity 59, # 101471 (2020)) the authors presented a family of points on the sphere \(\mathbb {S}^{2}\) depending on many parameters, called the Diamond ensemble. In this paper we compute the spherical cap discrepancy of the Diamond ensemble as well as some other quantities. We also define an area regular partition on the sphere where each region contains exactly one point of the set. For a concrete choice of parameters, we prove that the Diamond ensemble provides the best spherical cap discrepancy, known until now for a deterministic family of points.
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Acknowledgements
I would like to thank Peter Grabner for our discussions on the topic and for introducing me to the book [6], it was such a nice reading. I also want to thank Johann Brauchart for his corrections on the first version of this manuscript and the anonymous referees for their helpful comments.
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The author has been supported by the Austrian Science Fund FWF project F5503 (part of the Special Research Program (SFB) Quasi-Monte Carlo Methods: Theory and Applications), by MTM2017-83816-P from the Spanish Ministry of Science MICINN, and by 21.SI01.64658 from Universidad de Cantabria and Banco de Santander.
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Etayo, U. Spherical Cap Discrepancy of the Diamond Ensemble. Discrete Comput Geom 66, 1218–1238 (2021). https://doi.org/10.1007/s00454-021-00305-4
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DOI: https://doi.org/10.1007/s00454-021-00305-4