Discrepancies and their means

TL;DR

This paper presents an explicit formula for the discrepancy function of point distributions on a torus, linking it to mean values on sub-tori, and applies it to prove a theorem relating different discrepancy measures.

Contribution

It introduces a new explicit formula for discrepancy functions on tori and provides a simple proof of Lev's theorem on discrepancy equivalence.

Findings

Discrepancy function expressed via mean values on sub-tori

Proof of Lev's theorem on $L_{ ext{infty}}$- and shifted $L_q$-discrepancies

Simplification of discrepancy comparison methods

Abstract

It is shown that the discrepancy function for point distributions on a torus is expressed by an explicit formula in terms of its mean values on sub-tori. As an application of this formula, a simple proof of a theorem of Lev on the equivalence of $L_{\infty}$- and shifted $L_q$-discrepancies is given.