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How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ 1 result recorded, not proved here · all sources checked

Everything on this page is an external dependency: a result stated and cited to the literature, not proved anywhere in this library. There is no proof here to machine-check, audit or AI-judge, so this page carries none of those marks. What has been checked is what the page does assert — that each statement is correct, that its attribution and source are right, and that it is consistent with the rest of the library.

Optimal star-discrepancy order in dimension at least three

1 · Prerequisites

2 · Summary

Summary

Using the published half-open, equal-weight point-multiset definition of star discrepancy, this page asks for the exact minimum order when the integer dimension $d \geq 3$ is fixed before $n \to \infty$ . The candidate $n^{- 1} (\log n)^{d - 1}$ remains conjectural in this scope. Configurations are ordered $n$ -point lists, and repeated points count with multiplicity. The companion page is contextual and does not supply a proof.

3 · Logical flowchart

4 · Definitions, theorems and proofs

Open problemQuestion: Literature-sourcedProof: Not applicable‡ sources checked 2026-10-17‡ not proved hereOpen item page →

‡ Recorded, not proved here. This statement is included so the library can refer to it honestly, with a citation to the literature. It is not proved anywhere in this library: the track that would prove it has not been developed here yet.

Optimal star-discrepancy order at fixed dimension at least three

Open problem

Fix an integer $d \geq 3$ . For each integer $n \geq 2$ , 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 $δ_{d} (n) : = \inf {D_{n}^{*} (P) : P \in ([0, 1)^{d})^{n}},$ where $D_{n}^{*}$ is the normalized half-open anchored-box star discrepancy of Star discrepancy, digital nets, and digital sequences. Determine the exact asymptotic order of $δ_{d} (n)$ as $n \to \infty$ with $d$ fixed. In particular, prove or disprove the conjectured order $δ_{d} (n) ≍_{d} n^{- 1} (\log n)^{d - 1},$ and determine the correct order if this conjecture fails. Here $\log$ is the natural logarithm, and $≍_{d}$ means that positive constants $c_{d}, C_{d}$ and an integer $n_{0} (d)$ exist such that $c_{d} n^{- 1} (\log n)^{d - 1} \leq δ_{d} (n) \leq C_{d} n^{- 1} (\log n)^{d - 1}$ for all $n \geq n_{0} (d)$ .

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 $0 \leq D_{n}^{*} (P) \leq 1$ for every configuration. Thus $δ_{d} (n)$ is a well-defined finite real infimum; attainment is not asserted.

Status

Status reviewed on 2026-10-17: the inspected sources support retaining the fixed- $d \geq 3$ 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 $n^{- 1} (\log n)^{(d - 1) / 2}$ 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.

5 · Examples, counterexamples and false statements

None yet.