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The Kakeya maximal conjecture in three dimensions
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Abstract
We prove the Kakeya maximal conjecture in three dimensions. For every ε > 0, the maximal average over unit tubes of radius δ maps $L^3(\mathbb R^3)$ to $L^3(S^2)$ with norm at most $C_\varepsilon\delta^{-\varepsilon}$.
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Comments and discussion
- deffie (@df_163), post on X (2026-10-06). Points out that Wang and Zahl resolved the three-dimensional Kakeya set conjecture, while this claim concerns the stronger maximal conjecture, which implies it and which Wang and Zahl had not solved.
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)
- Cristian Rios, Eric T. Sawyer, Equivalence of linear and trilinear Kakeya conjectures in three dimensions (arXiv:2506.21315, v1 26 Jun 2025, v4 29 Sep 2025). This paper proves that the three-dimensional Kakeya maximal operator conjecture, which Theorem 1.1 claims to settle, is equivalent to a disjoint trilinear dual form, and treats the conjecture itself as open.
Digests and expositions
- OpenAI Math Digest, The Kakeya maximal conjecture in three dimensions (undated; accessed 2026-10-21). This explicitly AI-generated guide maps the claimed proof through critical exponents, stationary geometry, projection constraints, and the passage from shading estimates to arbitrary L³ functions. [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]
- 36Kr (credited to 新智元 / AI Era), Wang Hong Solves the 3D Kakeya Conjecture: OpenAI Releases 175-Page 4D Proof Manuscript (36Kr, 2026-10-10). This article describes the 97-page three-dimensional manuscript as aiming at a maximal-function statement stronger than the Wang–Zahl set theorem, and says both arguments "still need independent verification". [family-level]
Elsewhere
- VibeMathed: Problem record
- whataifound: Registry entry
- Emergent Mind: family 074
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026,
author = {{OpenAI}},
title = {{The Kakeya maximal conjecture in three dimensions}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf}{OAI:The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026}},
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:
- Every four-dimensional Kakeya set has full Hausdorff dimension · September 24, 2026