The Kakeya set conjecture in four dimensions (Hausdorff dimension)
A Kakeya set in contains a unit line segment in every direction. Besicovitch showed such a set can have measure zero; the Hausdorff-dimension Kakeya conjecture asserts it must still have Hausdorff dimension . It was known for (Davies, 1971) and (Wang and Zahl, 2025). In , Wolff's hairbrush gave dimension at least , and the best Hausdorff bound before this work was about (Katz and Zahl). Does every subset of containing a unit line segment in every direction have Hausdorff dimension ?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Harmonic analysis; geometric measure theory
- Posed by
- Hausdorff-dimension form of the Kakeya conjecture (classical, after Besicovitch and Davies); the manuscript cites the Katz-Tao (2002) formulation
- Year posed
- —
- Years open
- —
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 55 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: if contains, for every , a segment , then . No measurability, compactness, stickiness or packing-dimension hypothesis is imposed. The paper also derives packing and Minkowski-dimension consequences, a projection lower bound in higher dimensions, and Nikodym and curved-Kakeya applications via cited transfers. It does NOT prove the four-dimensional Kakeya maximal conjecture, and does not settle the conjecture in dimension five or higher.
What the AI did
The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. This family consists of two manuscripts dated September 23 and 24, 2026; this entry draws on the September 24 one. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the introduction states it; it covers arbitrary (not necessarily measurable or compact) sets and arbitrary witnessing segments, which is the full Hausdorff form in R^4. The proof was not refereed. No Lean formalization exists. It uses a weighted plank lemma (Lemma 2.3) from the companion three-dimensional maximal paper, itself unreviewed, together with the published Guth-Wang-Zahl union estimate.