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Every four-dimensional Kakeya set has full Hausdorff dimension
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Abstract
We prove the four-dimensional Hausdorff-dimension Kakeya conjecture: every subset of ℝ4 containing a unit line segment in every direction has Hausdorff dimension four. No compactness or regularity assumption is imposed on the set or its witnessing line family.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-23) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Florious (@Florious95), thread on X (2026-10-09). In Chinese, notes that this unreviewed four-dimensional proof takes the Wang–Zahl sticky Kakeya theorem and the Guth–Wang–Zahl three-dimensional estimates as key inputs, arguing it builds on that human work.
Formalization
- mo271, Wikipedia/Kakeya: the Kakeya set conjecture holds in dimension 4 (issue #6910) (google-deepmind/formal-conjectures, opened 2026-10-07). This open issue cites this paper and proposes adding
kakeya_4d : KakeyaSetConjectureDim 4, tagged as solved, toFormalConjectures/Wikipedia/Kakeya.lean; it contains no proof or linked PR and says the comparison was prepared with AI assistance.
Related prior literature mentioned by others
- Hong Wang and Joshua Zahl, Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions (arXiv:2502.17655v1, 2025-02-24). Azmat's Binary Verse AI exposition distinguishes this three-dimensional Hausdorff- and Minkowski-dimension theorem from the release's three-dimensional maximal and four-dimensional Hausdorff claims. [family-level]
- Larry Guth, Hong Wang, and Joshua Zahl, A streamlined proof of the Kakeya set conjecture in R³ (arXiv:2601.14411v1, 2026-01-20). Azmat's Binary Verse AI exposition identifies this union estimate as the foundation for the three-dimensional companion's weighted plank estimate used by the four-dimensional manuscript. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Mukul Rai Choudhuri, New bound on the Hausdorff dimension of Kakeya sets in $\mathbb{R}^4$ (arXiv:2610.12480, submitted 30 Aug 2026). This paper combines Wang–Zahl with the Katz–Zahl planebrush to give the lower bound 3.073, whereas this paper's introduction names 3.059 as the prior Hausdorff bound.
- Tiago Moreira, Refinements on the Restriction and Kakeya problems range in $\mathbb{R}^4$ (arXiv:2610.06181, 5 Oct 2026). This paper, posted after this manuscript's date, raises the four-dimensional Kakeya maximal dimension parameter to about 3.06737, above the value 3.0543 (Borges et al.) that the introduction treats as current.
- Joshua Zahl, A survey of the Kakeya conjecture, 2000–2025 (arXiv:2512.09397, Dec 2025). This survey covers the status of the Kakeya problem in dimensions four and higher before this paper; the companion three-dimensional paper cites it, but this paper's bibliography does not.
Digests and expositions
- OpenAI Math Digest, Every four-dimensional Kakeya set has full Hausdorff dimension (undated; accessed 2026-10-21). This explicitly AI-generated guide maps the reduction from Hausdorff covers to positive horizon and the three narrowness regimes, and flags a numbering discrepancy in the citation of the companion's weighted plank estimate. [AI-generated per source]
News and blogs
- Azmat, Kakeya Conjecture: How OpenAI Claims to Break the 4D Barrier (Binary Verse AI, 2026-10-07). This exposition discusses both manuscripts, distinguishes their claims from the Wang–Zahl set theorem, and outlines the four-dimensional argument and its dependence on the three-dimensional companion. [family-level]
Elsewhere
- VibeMathed: Problem record
- whataifound: Registry entry
- Emergent Mind: family 074
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026,
author = {{OpenAI}},
title = {{Every Four-Dimensional Kakeya Set Has Full Hausdorff Dimension}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf}{OAI:Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026}},
year = {2026}
}Family 074: Kakeya in three and four dimensions
Resolves the Kakeya maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four. In three dimensions, the radius-δ tube maximal operator maps $L^3(\mathbb R^3)$ to $L^3(S^2)$ with norm $O_\varepsilon(\delta^{-\varepsilon})$ for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.
Other papers in this family:
- The Kakeya maximal conjecture in three dimensions · September 23, 2026