Optimal star-discrepancy order at fixed dimension at least three

Fix an integer $d\ge 3$. For each integer $n\ge 2$, let $P=(x_1,\dots ,x_n)$ range over the ordered configurations in $([0,1{)}^d{)}^n$, with repeated points counted with multiplicity, and set ${\delta }_d(n):=\inf \{D_n^{\ast }(P):P\in ([0,1{)}^d{)}^n\},$ where $D_n^{\ast }$ is the normalized half-open anchored-box star discrepancy of Star discrepancy, digital nets, and digital sequences. Determine the exact asymptotic order of ${\delta }_d(n)$ as $n\to \mathrm{\infty }$ with $d$ fixed. In particular, prove or disprove the conjectured order ${\delta }_d(n){\asymp }_d{n}^{-1}(\log n{)}^{d-1},$ and determine the correct order if this conjecture fails. Here $\log$ is the natural logarithm, and ${\asymp }_d$ means that positive constants $c_d,C_d$ and an integer $n_0(d)$ exist such that $c_d{n}^{-1}(\log n{)}^{d-1}\le {\delta }_d(n)\le C_d{n}^{-1}(\log n{)}^{d-1}$ for all $n\ge n_0(d)$.

The configuration set is nonempty because it contains the list with every point equal to $0\in [0,1{)}^d$. Each empirical fraction and box volume lies in $[0,1]$, so $0\le D_n^{\ast }(P)\le 1$ for every configuration. Thus ${\delta }_d(n)$ is a well-defined finite real infimum; attainment is not asserted.

Status reviewed on 2026-10-17: the inspected sources support retaining the fixed-$d\ge 3$ question as open. Owen's Practical Quasi-Monte Carlo, printed pp. 18–19, says the best possible rate is not known, records the classical Roth lower baseline $n^{-1}(\log n{)}^{(d-1)\mathrm{/}2}$ and the conjectural $n^{-1}(\log n{)}^{d-1}$ rate, and distinguishes the settled planar order $n^{-1}\log n$. The Roth bound is stated here only as a classical baseline, not as the strongest known lower bound. Owen's Theorem 15.7, printed p. 27, supplies an upper-order construction.

Dick's version 3 paper, dated 2026-10-15, states a result on uniform polynomial tractability exponents for inverse star discrepancy. Its Remark 1.4 explicitly separates that growing-dimension result from each fixed-dimension ray and gives a fixed-d construction upper bound; it does not state a matching fixed-d lower bound. The refreshed bounded current-results review recorded in the batch coverage and the ROOT cycle-1 report found no inspected source resolving this exact question. That review is not exhaustive and does not prove that no resolution exists. Publication requires a fresh source/version/status check. No proof of the conjectured lower bound is claimed, and this open problem is never a logical supplier.