Improve star discrepancy bounds in dimensions d > 2

Improve the best-known upper bounds for the star discrepancy D^*(X) of n-point sets X ⊂ [0,1]^d when d > 2, where D^*(X) is the maximum continuous discrepancy over corners (axis-parallel boxes anchored at the origin); specifically, determine tighter asymptotic bounds than the current O(log^{d−1} n) upper bound for d > 2.

Background

Star discrepancy D*(X) is a central measure in quasi-Monte Carlo theory, directly controlling numerical integration error via Koksma–Hlawka-type inequalities. Many explicit constructions of low-discrepancy sets are known, with tight bounds for certain discrepancy measures.

For D*(X), the current best-known upper bound is O(log{d−1} n), and for d = 2 there is a tight lower bound of Ω(log n). The longstanding goal is to improve the bounds for d > 2, a problem highlighted in the discrepancy literature and restated here as a great open problem.

References

It remains a 'great open problem' [BC87] to improve the bound on $D*(X)$ for $d > 2$ (for $d=2$, a tight lower bound of $\Omega(\log n)$ is known).

— Quasi-Monte Carlo Beyond Hardy-Krause  (2408.06475 - Bansal et al., 2024) in Appendix: Further Related Work

While it is still unclear how to analyze their proposed greedy strategy, we give a new variant of Haar-thinning that we call {\em linear-feedback Haar-thinning} which attains their conjectured $O_{d,\varepsilon}(\log{d+1} n)$ bound.

— Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice  (2609.10921 - Ezeunala et al., 10 Sep 2026) in Section 1, subsection “Our Results,” paragraph “Low-Discrepancy Sequences via Online Thinning”

Interestingly, the optimal star discrepancy bound for point sets and sequences remains unknown when $d \geq 3$.

— Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice  (2609.10921 - Ezeunala et al., 10 Sep 2026) in Section 1, subsection “Further Related Work,” paragraph “Low-Discrepancy Point Sets and Sequences”