On the lower bounds for the spherical cap discrepancy

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 $\mathbb S^4$, $\mathbb S^8$, $\mathbb S^{24}$).