Well-distributed great circles on S^2

TL;DR

This paper investigates the overlap properties of neighborhoods around great circles on the sphere, establishing sharp bounds and exploring connections to energy minimization problems like the Thomson problem.

Contribution

It provides sharp bounds on overlaps of neighborhoods of great circles on the sphere and links these results to minimal energy configurations such as the Thomson problem.

Findings

Established bounds for overlaps depending on parameter s

Identified arrangements with bounded overlaps for s=1

Connected geometric overlap problems to energy minimization on spheres

Abstract

Let $C_1, \dots, C_n$ denote the $1/n-$neighborhood of $n$ great circles on $\mathbb{S}^2$. We are interested in how much these areas have to overlap and prove the sharp bounds $$ \sum_{i, j = 1 \atop i \neq j}^{n}{|C_i \cap C_j|^s} \gtrsim_s \begin{cases} n^{2 - 2s} \qquad &\mbox{if}~0 \leq s < 2 \\ n^{-2} \log{n} \qquad &\mbox{if}~s = 2\\ n^{1- 3s/2} \qquad &\mbox{if}~s > 2. \end{cases} .$$ For $s=1$ there are arrangements for which the sum of mutual overlap is uniformly bounded (for the analogous problem in $\mathbb{R}^2$ the lower bound is $\gtrsim \log{n}$) and there are strong connections to minimal energy configurations of $n$ charged electrons on $\mathbb{S}^2$ (the J. J. Thomson problem).