Irregularities of distribution on two point homogeneous spaces

TL;DR

This paper investigates the irregularities of distribution on two-point homogeneous spaces, providing sharp estimates for discrepancy measures related to point distributions and their deviations from uniformity.

Contribution

It establishes new lower bounds for discrepancy integrals on two-point homogeneous spaces, extending understanding of distribution irregularities in these geometric settings.

Findings

Derived sharp discrepancy bounds for various radii

Extended irregularity estimates to higher-dimensional spaces

Provided explicit constants for discrepancy lower bounds

Abstract

We study the irregularities of distribution on two-point homogeneous spaces. Our main result is the following: let $d$ be the real dimension of a two point homogeneous space $\mathcal{M}$, let $\left( \{ a_{j}\} _{j=1}^{N},\{ x_{j}\} _{j=1}^{N}\right) $ be a system of positive weights and points on $\mathcal{M}$ and let \[ D_{r}( x) =\sum_{j=1}^{N}a_{j}\chi_{B_{r}(x)}(x_{j})-\mu(B_{r}(x)) \] be the discrepancy associated with the ball $B_{r}( x) $. Then, if $d\not \equiv 1(\operatorname{mod}4)$, for any radius $0<r<\pi/2$, we obtain the sharp estimate \[ \int_{\mathcal{M}}\left( \left\vert D_{r}( x) \right\vert ^{2}+\left\vert D_{2r}( x) \right\vert ^{2}\right) d\mu( x) \geqslant cN^{-1-\frac{1}{d}}. \]