Low energy points on the sphere and the real projective plane

TL;DR

This paper generalizes a point configuration on the sphere to the real projective plane, providing bounds for energy minimization, which advances understanding of optimal point distributions on these surfaces.

Contribution

It extends the Diamond ensemble construction from the sphere to the real projective plane, offering explicit energy bounds and new insights into point configurations on these surfaces.

Findings

Constructed point sets with low energy on the real projective plane.

Derived explicit upper and lower bounds for Green and logarithmic energy.

Demonstrated the effectiveness of the generalized ensemble in energy minimization.

Abstract

We present a generalization of a family of points on $\mathbb{S}^2$, the Diamond ensemble, containing collections of $N$ points on $\mathbb{S}^2$ with very small logarithmic energy for all $N\in\mathbb{N}$. We extend this construction to the real projective plane $\mathbb{RP}^2$ and we obtain upper and lower bounds with explicit constants for the Green and logarithmic energy on this last space.