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Optimal star-discrepancy order in dimension at least three — Examples

1 · Prerequisites

2 · Summary

Summary

The example verifies the exact finite discrepancy of the center point in dimensions two and three from Star discrepancy, digital nets, and digital sequences. For literature context, Owen reports the planar order $n^{- 1} \log n$ and, for fixed $d \geq 3$ , a classical Roth lower baseline $n^{- 1} (\log n)^{(d - 1) / 2}$ alongside a construction upper order $n^{- 1} (\log n)^{d - 1}$ . The lower rate is only a classical baseline here, not a claim of best-known status. Those exponents differ for $d \geq 3$ , so these bounds do not establish the conjectured order. Dick's version 3 result concerns uniform growing-dimension tractability; its Remark 1.4 distinguishes fixed $d$ . This page makes no resolution or counterexample claim. Owen, Chapter 15, printed pp. 17–19 and Theorem 15.7; Dick, arXiv:2607.23571v3, §1 and Remark 1.4.

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Exact star discrepancy of the center point

Example

For an integer $d \geq 2$ , let $P = (x)$ be the singleton configuration with $x = (1 / 2, \dots, 1 / 2) \in [0, 1)^{d}$ . Its star discrepancy is $D_{1}^{*} (P) = 1 - 2^{- d}$ . Thus the exact values are $3 / 4$ in dimension two and $7 / 8$ in dimension three.

Facts & Assumptions

Given: An integer $d \geq 2$ and the singleton configuration $P = (1 / 2, \dots, 1 / 2)$ . Compute $D_{1}^{*} (P)$ using the half-open anchored-box convention.

[F1]

Star discrepancy, digital nets, and digital sequences: A point multiset is an ordered list in $[0, 1)^{d}$ , repetitions are counted with multiplicity, and $D_{N}^{*}$ is the supremum over $u \in [0, 1]^{d}$ of the absolute difference between the fraction of listed points in $\prod_{j} [0, u_{j})$ and its volume $\prod_{j} u_{j}$ .

Verification

1.1givenF1

For $u \in [0, 1]^{d}$ , the point $x$ lies in $\prod_{j = 1}^{d} [0, u_{j})$ exactly when $u_{j} > 1 / 2$ for every $j$ , because the box is half-open. The point count is therefore one if all these inequalities hold and zero otherwise.

2.1givenstep 1.1algebra

If some $u_{j} \leq 1 / 2$ , the count is zero, so the discrepancy is $\prod_{j} u_{j} \leq u_{j} \leq 1 / 2$ . Since $d \geq 2$ , $1 - 2^{- d} \geq 3 / 4 > 1 / 2$ , so this case contributes less than $1 - 2^{- d}$ .

2.2givenstep 1.1algebra

If every $u_{j} > 1 / 2$ , the count is one and the discrepancy is $1 - \prod_{j} u_{j}$ . The product exceeds $2^{- d}$ , so this value is less than $1 - 2^{- d}$ .

3.1givenF1step 2.1step 2.2choosealgebra∎

For each integer $k \geq 3$ , the box endpoint $u^{(k)} = (1 / 2 + 1 / k, \dots, 1 / 2 + 1 / k)$ lies in $[0, 1]^{d}$ and has discrepancy $1 - (1 / 2 + 1 / k)^{d} \to 1 - 2^{- d}$ . Thus the supremum is at least $1 - 2^{- d}$ , while steps 2.1–2.2 bound every box error above by this value; hence $D_{1}^{*} (P) = 1 - 2^{- d}$ . Substitution gives $D_{1}^{*} (P) = 3 / 4$ for $d = 2$ and $7 / 8$ for $d = 3$ .

Remarks

The finite calculation is proved only from the definition; it is not an asymptotic-rate claim. For context, Owen reports the planar order $n^{- 1} \log n$ and, for fixed $d \geq 3$ , a classical Roth lower baseline $n^{- 1} (\log n)^{(d - 1) / 2}$ alongside a known construction upper order $n^{- 1} (\log n)^{d - 1}$ . The cited lower bound is not presented as the strongest known lower bound. Dick's version 3 result concerns uniform growing-dimension tractability, and Remark 1.4 keeps fixed dimension separate. These source-reported comparisons do not prove a matching lower bound or resolve the open question.