On the lower bounds for the spherical cap discrepancy

TL;DR

This paper presents a simple proof of a classical lower bound on spherical cap discrepancy for points on spheres, introduces new estimates related to geometric quantities, and refines bounds with explicit constants close to conjectured optimal values.

Contribution

It offers a new elementary proof of the spherical cap discrepancy lower bound and derives several new geometric and discrepancy estimates with explicit constants.

Findings

Lower bound of discrepancy is of order N^{-1/2 - 1/(2d)}

Explicit constants in discrepancy bounds are within 3-7% of conjectured optimal values

New estimates relate discrepancy to geometric quantities and distances

Abstract

We start by providing a very simple and elementary new proof of the classical bound due to J. Beck which states that the spherical cap $\mathbb{L}_2$-discrepancy of any $N$ points on the unit sphere $\mathbb S^d$ in $\mathbb{R}^{d+1}$, $d\geq2$, is at least of the order $N^{-\frac12-\frac{1}{2d}}$. The argument used in this proof leads us to many further new results: estimates of the discrepancy in terms of various geometric quantities, an easy proof of {point-independent} upper estimates for the sum of positive powers of Euclidean distances between points on the sphere, lower bounds for the discrepancy of rectifiable curves and sets of arbitrary Hausdorff dimension. Moreover, refinements of the proof also allow us to obtain explicit values of the constants in the lower discrepancy bound on $\mathbb{S}^d$. The value of the obtained asymptotic constant falls within $3\%$ of the conjectured optimal constant on $\mathbb S^2$ (and within up to $7\%$ on $\mathbb S^4$, $\mathbb S^8$, $\mathbb S^{24}$).