Quasi-uniform digital nets in arbitrary dimension
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Can explicit low-discrepancy digital nets also have both point separation and covering radius of order N^(−1/s) in every fixed dimension s≥3? Separation prevents clustering; covering radius controls empty regions.
Why it matters and what counts as progress
Why it matters: The combination asks for good integration point sets with uniform geometric spacing.
What counts as progress: An explicit net family meeting both geometric bounds under stated base conditions, while retaining the digital-net discrepancy property.
How the problem developed
Josef Dick, Takashi Goda and Kosuke Suzuki Studied quasi-uniformity of digital nets, giving a low-discrepancy quasi-uniform example in dimension two and counterexamples for other familiar nets.
Leonhard Gruenschloss, with Claude Opus 5.5 and earlier Claude Fable 5.1 Released explicit low-discrepancy quasi-uniform digital nets in every dimension s≥3. The construction and Lean formalization were produced by AI systems under the author’s direction; the separation bound requires an additional base threshold.
AI contribution
Claude agents construct quasi-uniform digital nets in every dimension
For every s≥3, allowed base q and k≥1, the explicit N=q^(sk) digital net has covering radius ≤√s N^(−1/s); when q≥5a*_s+4 its point separation is ≥(1−a*_s/q)N^(−1/s), with the stated bound on a*_s.
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Claude agents construct quasi-uniform digital nets in every dimension