Tusnády’s problem claimed settled twice, with AI help
Awaiting review
Importance: major (4 of 5)The takeaway
On 6 Nov 2026 arXiv announced two independent preprints, by Nicola Wengiel and by Yuhao Lei, each proving that the worst-case combinatorial discrepancy of N points with respect to axis-parallel boxes in fixed dimension d is Θ((log N)^(d−1/2)), matching Nikolov’s upper bound.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 6 November 2026
Your AI and this story
- GPT-6 Astra190 days after its cutoff
- Claude Opus 5.5129 days after its cutoff
- Gemini 4 Argon220 days after its cutoff
- Grok 4.7159 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 129 days before it.
Key facts
- Wengiel, ‘Resolving Tusnády’s Problem’ (arXiv 2611.06998): lower bound Ω_d((log N)^(d−1/2)) for anchored boxes, every d ≥ 3; with Nikolov’s upper bound and Wei’s planar lower bound gives Θ_d((log N)^(d−1/2)) for all d ≥ 2
- Wengiel’s acknowledgement: ‘The author used Anthropic’s Claude and OpenAI’s Codex to assist with literature research, proof development and checking, and manuscript preparation.’
- Lei, ‘Tusnády’s problem and online two-sample thinning’ (arXiv 2611.08455): same sharp order in every d ≥ 2, plus a robust version (holds for every subset after deleting a small fixed fraction of points)
- Lei’s disclosure: ‘ChatGPT was used to generate ideas, assist with the proof, review draft, check main argument, revise the manuscript and preparation of the mathematical exposition.’
- Earlier status: the gap between lower bounds of order (log N)^(d−1) and Nikolov’s (log N)^(d−1/2) upper bound had stood for d ≥ 3; the planar case was settled by Wei
What happened
Two preprints in the 6 Nov 2026 arXiv announcement claim the sharp order of discrepancy for axis-parallel boxes in every fixed dimension: arXiv 2611.069981 (Nicola Wengiel, math.CA) and arXiv 2611.084552 (Yuhao Lei, math.CO). Both state that the largest combinatorial discrepancy is Θ_d((log N)^(d−1/2)), matching Nikolov’s upper bound. Wengiel’s proof uses smoothed product test functions; Lei’s goes through an online two-sample thinning argument and gives a stronger, robust statement. The AI statements are quoted above from the PDFs, as read by the sweep’s disclosure scan.
Why it matters
Tusnády’s problem is a standard open problem of geometric discrepancy theory, and the gap in dimensions d ≥ 3 had been open for decades. Two independent claims on the same day make cross-checking easier, but neither has been refereed and no expert comment was found on 6 Nov (Hacker News search was unavailable during this run).
What is disputed or not yet verified
| Verification | Two unrefereed arXiv preprints; no Lean formalization stated |
|---|
Sources
2 sources from 1 site. Numbers match the chips in the text.
2 sources: 2 primary
Primary
- Wengiel, Resolving Tusnády’s Problem (arXiv 2611.06998)arxiv.org, paper
- Lei, Tusnády’s problem and online two-sample thinning (arXiv 2611.08455)arxiv.org, paper
Changes
- Filed from the arXiv announcement of 6 Nov