Single radius spherical cap discrepancy on compact two-point homogeneous spaces

TL;DR

This paper investigates lower bounds for the discrepancy of spherical caps of fixed radius on compact two-point homogeneous spaces, revealing dimension-dependent conditions and limitations for such estimates.

Contribution

It provides new lower bound estimates for spherical cap discrepancy on these spaces, including conditions on the radius and dimension-specific limitations.

Findings

Established lower bounds for discrepancy integral involving point sets.

Identified dimension-dependent conditions on the radius for discrepancy estimates.

Proved weaker bounds for certain dimensions where conditions are not met.

Abstract

In this note we study estimates from below of the single radius spherical discrepancy in the setting of compact two-point homogeneous spaces. Namely, given a $d$-dimensional manifold $\mathcal M$ endowed with a distance $\rho$ so that $(\mathcal M, \rho)$ is a two-point homogeneous space and with the Riemannian measure $\mu$, we provide conditions on $r$ such that if $D_r$ denotes the discrepancy of the ball of radius $r$, then, for an absolute constant $C>0$ and for every set of points $\{x_j\}_{j=1}^N$, one has $\int_{\mathcal M} |D_{r}(x)|^2\, d\mu(x)\geqslant C N^{-1-\frac1d}$. The conditions on $r$ that we have depend on the dimension $d$ of the manifold and cannot be achieved when $d \equiv 1 \ ( \operatorname{mod}4)$. Nonetheless, we prove a weaker estimate for such dimensions as well.