The BMO-discrepancy suffers from the curse of dimensionality

TL;DR

This paper demonstrates that the minimal discrepancy of point sets in high-dimensional cubes, measured by the BMO seminorm, deteriorates exponentially with increasing dimension, highlighting a significant challenge in high-dimensional discrepancy theory.

Contribution

The paper establishes that the BMO-discrepancy suffers from the curse of dimensionality, providing a fundamental limitation in high-dimensional discrepancy measures.

Findings

BMO-discrepancy increases exponentially with dimension

High-dimensional discrepancy measures face fundamental limitations

Results highlight challenges in high-dimensional uniformity assessment

Abstract

We show that the minimal discrepancy of a point set in the $d$-dimensional unit cube with respect to the BMO seminorm suffers from the curse of dimensionality.