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.