Laczkovich’s Lipschitz mapping problem: claimed answer, ChatGPT key step
Awaiting review
Importance: major (4 of 5)The takeaway
In 1991 Miklós Laczkovich asked whether every measurable set of positive measure in ℝ^d can be mapped onto a cube by a Lipschitz map. Preiss and Matoušek answered it in the plane; dimension 3 and up stayed open. On 28 Oct Attila Gáspár posted a positive answer in every dimension. His disclosure says the proof of Theorem 1.8 ‘was obtained using ChatGPT on 26 October’, and Proposition 2.8 with Lemma 2.7 is ‘entirely due to ChatGPT’. He had known other parts and the lower bound since early 2026. It is a short preprint, not refereed.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 30 October 2026
Your AI and this story
- GPT-6 Astra181 days after its cutoff
- Claude Opus 5.5120 days after its cutoff
- Gemini 4 Argon211 days after its cutoff
- Grok 4.7150 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 120 days before it.
Key facts
- Claim (abstract): ‘In 1991, Laczkovich asked whether every measurable set of positive measure in ℝ^d can be mapped onto a cube with a Lipschitz map. We show that the answer is positive for every d.’
- AI disclosure (verbatim): ‘The proof of Theorem 1.8 was obtained using ChatGPT on 26 October. In particular, the proof of Proposition 2.8, including the statement and proof of Lemma 2.7, is entirely due to ChatGPT. This proof was rewritten by the author to make it easier to understand. The proof of Proposition 2.9 was also written with the help of ChatGPT.’ The lower bound for γ(C) (Prop. 2.5) and the consequences of equality in Theorem 1.8 were the author’s own, from early 2026, without AI
- On the model’s limits (verbatim): ‘When asked to give lower bounds for the value of γ for specific sets, ChatGPT did not come up with this lower bound, though it could prove it given the statement.’
- Prior state (paper): d = 1 is elementary; d = 2 by Preiss, then Matoušek (an absolute constant c with a 1-Lipschitz surjection A → [0, c]² when λ(A) = 1, via Erdős–Szekeres) and Jones (via Uy’s theorem); ‘until now, the case d ≥ 3 was open’. The proof uses the author’s theory of generalized strips (proximal maps of convex functions)
- Also answers Question 4.2 of an earlier paper: an ε-perturbation of the identity maps A into finitely many hyperplanes; and Kolmogorov’s problem for Jordan measurable sets
- Verification status: arXiv preprint, posted two days after the AI step; not refereed, no formal proof; no expert reaction or HN thread found on 30 Oct
What happened
Miklós Laczkovich asked in 1991 whether any measurable set of positive volume can be squeezed by a Lipschitz map onto a full cube. The answer was known to be yes in the plane, through Preiss and later Matoušek, but open in three dimensions and above.
Attila Gáspár posted a proof for every dimension on 28 Oct. He states that the main inequality’s proof came from ChatGPT on 26 Oct and that one proposition and its lemma are entirely the model’s; he rewrote them and supplied the lower bound and the framework himself.
Why it matters
It is a classic question of geometric measure theory with a clear statement, and the paper separates the AI’s step from the author’s earlier work in unusual detail, including what the model failed to find. It was posted two days after the AI step and still needs checking.
What is disputed or not yet verified
| Verification | Preprint; not refereed |
|---|
Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
Changes
- Filed (science agent; arXiv abstract and PDF read, AI disclosure quoted)