Asymptotic Properties of Discrete Minimal $s,lo{g}^t$-Energy Constants and Configurations

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 $s\to \infty$ and as $s\to s_0>0,$ 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.