Discrepancy of second order digital sequences in function spaces with dominating mixed smoothness

TL;DR

This paper investigates the discrepancy of second order digital sequences in various function spaces with dominating mixed smoothness, providing sharp bounds for their deviation from uniformity.

Contribution

It introduces new bounds for the discrepancy function of digital sequences in advanced function spaces, extending classical results to more general norms.

Findings

Sharp bounds for discrepancy in BMO and Orlicz norms

Results applicable to Sobolev, Besov, Triebel-Lizorkin spaces

Enhanced understanding of distribution uniformity in high-dimensional spaces

Abstract

The discrepancy function measures the deviation of the empirical distribution of a point set in $[0,1]^d$ from the uniform distribution. In this paper, we study the classical discrepancy function with respect to the BMO and exponential Orlicz norms, as well as Sobolev, Besov and Triebel-Lizorkin norms with dominating mixed smoothness. We give sharp bounds for the discrepancy function under such norms with respect to infinite sequences.