Hammersley Point Sets and Inverse of Star-Discrepancy

TL;DR

This paper improves the upper bounds on the star-discrepancy of N-point sets in high dimensions by combining recent theoretical results, leading to tighter discrepancy bounds than previously known.

Contribution

It establishes new, improved bounds on star-discrepancy for N-point sets in high dimensions using a novel combination of recent discrepancy and bracketing number results.

Findings

New upper bound on star-discrepancy: 2.4631832 * sqrt(d/N)

Improved discrepancy bounds for Hammersley point sets in dimensions 1 to 4

Enhanced understanding of discrepancy behavior in high-dimensional settings

Abstract

We establish the existence of $N$-point sets in dimension $d$ whose star-discrepancy is bounded above by $2.4631832 \sqrt{\frac{d}{N}}$, where the numerical constant improves upon all previously known bounds. This improvement is obtained by combining a recent result by Gnewuch on bracketing numbers in high dimensions with discrepancy bounds for Hammersley point sets due to Atanassov in dimensions $1 \leq d \leq 4$.