Lower and upper bounds for configurations of points on a sphere

TL;DR

This paper introduces new spectral decomposition-based proofs for bounds on sums involving point configurations on spheres, with applications to the Thomson problem and point distributions in 3D volumes.

Contribution

It provides novel spectral decomposition proofs for bounds on frame potentials and related sums, extending previous results and deriving sharp bounds in three dimensions.

Findings

Spectral decomposition proof of Sidelnikov's bound

New bounds for weighted sums of point configurations

Sharp upper bound for a 3D triple sum involving cross products

Abstract

We present a new proof (based on spectral decomposition) of a bound originally proved by Sidelnikov~\, for the frame potentials $\sum_{ij} \left( {\bf P}_i \cdot {\bf P}_j \right)^\ell $ on a unit--sphere in $d$ dimensions. Sidelnikov's bound is a special case of the lower bound for the weighted sums $\sum_{ij} f_i f_j \left( {\bf P}_i \cdot {\bf P}_j \right)^\ell$, where $f_i>0$ are scalar quantities associated to each point on the sphere, which we also prove using spectral decomposition. Moreover, in three dimensions, again using spectral decomposition, we find a sharp upper bound for $\sum_{ijk}^N \left[ \left( {\bf P}_i \times {\bf P}_j\right) \cdot {\bf P}_k \right]^2$. We explore two applications of these bounds: first, we examine configurations of points corresponding to the local minima of the Thomson problem for $N=972$; second, we analyze various distributions of points within a three-dimensional volume, where a suitable weighted sum is defined to satisfy a specific bound.