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arXiv · 2604.14326

Energy, Polarization, and Separation of Greedy Sequences for Riesz and Green Kernels

Dmitriy Bilyk·Liudmyla Kryvonos·Ryan W. Matzke·Edward Saff
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Abstract

We investigate the asymptotic behavior of greedy $s$-Riesz and Green energy sequences $\{x_{n}\}_{n=1}^{\infty}$ on the unit sphere $\mathbb{S}^{d} \subset \mathbb{R}^{d+1}$, where each point $x_n$ is defined as the minimizer of the discrete potential generated by the preceding points $x_1, x_2, ..., x_{n-1}$. We show that the greedy sequence attains optimal growth behavior for the second-order term of the Green and Riesz $s$-energies when $d-2 \leq s < d$. The main idea is to establish the bounds on polarization using well-separation properties of the greedy configurations.

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Dmitriy Bilyk, Liudmyla Kryvonos, Ryan W. Matzke, Edward Saff. 2026-04-15. Energy, Polarization, and Separation of Greedy Sequences for Riesz and Green Kernels. https://arxiv.org/abs/2604.14326

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