Bounds for $L_p$-discrepancies of point distributions in compact metric spaces

TL;DR

This paper extends upper bounds for $L_p$-discrepancies of point distributions from specific spaces to more general compact metric measure spaces, under simple volume conditions.

Contribution

It generalizes previous bounds on $L_p$-discrepancies to broader metric spaces with minimal assumptions on volume growth.

Findings

Bounds are applicable to a wider class of metric spaces.

The results hold for all $0<p\,\le\,\infty$.

Simplifies conditions needed for discrepancy bounds.

Abstract

Upper bounds for the $L_p$-discrepancies of point distributions in compact metric measure spaces for $0<p\le\infty$ have been established in the paper [6] by Brandolini, Chen, Colzani, Gigante and Travaglini. In the present paper we show that such bounds can be established in a much more general situation under very simple conditions on the volume of metric balls as a function of radii.