Hammersley Point Sets and Inverse of Star-Discrepancy

TL;DR

This work advances the inverse star-discrepancy problem by proving the existence of $N$-point sets in dimension $d$ with $D^*(N,d) \le 2.4631832 \sqrt{d/N}$, and consequently $N^*(\varepsilon,d) \le 6.0672715\, d\, \varepsilon^{-2}$. The key methodology combines a sharp bracketing-number bound from \citet{Gnewuch2024} with discrepancy guarantees for Hammersley point sets due to Atanassov, together with a lifting construction from Halton sequences. Small dimensions ($d \le 4$) are addressed via Atanassov bounds and targeted computer verification of finite exceptional cases, while a probabilistic/bracketing framework handles larger $d$, aided by a triangle-inequality reduction to minimize computations. The results provide tighter, practically relevant estimates for the inverse star-discrepancy and improve understanding of high-dimensional quasi-Monte Carlo sampling for numerical integration.

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 $1 \leq d \leq 4$.

Theorems & Definitions (9)

• Theorem 1
• Lemma 2: Gnewuch, Gnewuch2024, Theorem 2.9
• Lemma 3
• proof
• Lemma 4: GPW21
• Lemma 5: Atanassov, Ata04
• proof : Proof of Theorem \ref{['thm:improved:c2']}
• Remark 6
• Corollary 7