Asymptotic Properties of Discrete Minimal $s,log^t$-Energy Constants and Configurations

TL;DR

This paper studies the asymptotic behavior of minimal $s, ext{log}^t$-energy configurations on compact metric spaces, revealing their limits and optimal arrangements as parameters vary, and establishing properties of the energy constants.

Contribution

It introduces the $s, ext{log}^t$-energy concept and analyzes the asymptotic properties and limits of minimal energy configurations, including their convergence to best-packing arrangements.

Findings

Configurations tend to best-packing as $s o \infty$.

Configurations tend to $s_0, ext{log}^t$-energy configurations as $s o s_0 > 0$.

Optimality of equally spaced points on circles for certain energy problems.

Abstract

Combining the ideas of Riesz $s$-energy and $\log$-energy, we introduce the so-called $s,\log^t$-energy. In this paper, we investigate the asymptotic behaviors for $N,t$ fixed and $s$ varying of minimal $N$-point $s,\log^t$-energy constants and configurations of an infinite compact metric space of diameter less than $1$. In particular, we study certain continuity and differentiability properties of minimal $N$-point $s,\log^t$-energy constants in the variable $s$ and we show that in the limits as $s\rightarrow \infty$ and as $s\rightarrow s_0>0,$ minimal $N$-point $s,\log^t$-energy configurations tend to an $N$-point best-packing configuration and a minimal $N$-point $s_0,\log^t$-energy configuration, respectively. Furthermore, the optimality of $N$ distinct equally spaced points on circles in $\mathbb{R}^2$ for some certain $s,\log^t$ energy problems was proved.