Dyadic shift randomization in classical discrepancy theory

TL;DR

This paper investigates dyadic shift randomizations of point distributions in multi-dimensional spaces, providing explicit discrepancy formulas and sharp bounds using Rademacher functions and probabilistic inequalities.

Contribution

It introduces explicit discrepancy formulas for dyadic shifts and derives sharp bounds using probabilistic tools, advancing classical discrepancy theory.

Findings

Explicit formulas for discrepancies in terms of Rademacher functions

Sharp upper and lower bounds for mean discrepancies

Application of probabilistic inequalities to discrepancy analysis

Abstract

Dyadic shifts of point distributions in the multi-dimensional unit cube are considered as a randomization. Explicit formulas for the discrepancies of such randomized distributions are given in the paper in terms of Rademacher functions. Relaying on the statistical independence of Rademacher functions, Khinchin's inequalities, and other related results, we obtain very sharp upper and lower bounds for the mean discrepancies.