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arXiv · 2302.13067

Polarization and Greedy Energy on the Sphere

Dmitriy Bilyk·Michelle Mastrianni·Ryan W. Matzke·Stefan Steinerberger
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Abstract

We investigate the behavior of a greedy sequence on the sphere $\mathbb{S}^d$ defined so that at each step the point that minimizes the Riesz $s$-energy is added to the existing set of points. We show that for $0<s<d$, the greedy sequence achieves optimal second-order behavior for the Riesz $s$-energy (up to constants). In order to obtain this result, we prove that the second-order term of the maximal polarization with Riesz $s$-kernels is of order $N^{s/d}$ in the same range $0<s<d$. Furthermore, using the Stolarsky principle relating the $L^2$-discrepancy of a point set with the pairwise sum of distances (Riesz energy with $s=-1$), we also obtain a simple upper bound on the $L^2$-spherical cap discrepancy of the greedy sequence and give numerical examples that indicate that the true discrepancy is much lower.

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Dmitriy Bilyk, Michelle Mastrianni, Ryan W. Matzke, Stefan Steinerberger. 2023-07-30. Polarization and Greedy Energy on the Sphere. https://arxiv.org/abs/2302.13067

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