The Kakeya conjecture in three and four dimensions

The Kakeya conjecture asks how small a set can be that contains a unit line segment in every direction. Its Hausdorff-dimension form predicts that every such set in Rⁿ has Hausdorff dimension n, and the stronger maximal form predicts an Lⁿ bound with arbitrarily small loss for the Kakeya maximal operator. Two manuscripts dated 23 and 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim the maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four dimensions. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

• As recounted in the introductions of the manuscripts:
  • Besicovitch showed that a set containing a segment in every direction can have measure zero, and Davies proved full Hausdorff dimension in the plane; Córdoba gave planar maximal estimates. Wolff's hairbrush argument gave the bound (n + 2)/2, and Katz, Łaba and Tao and later Katz and Zahl improved it in three dimensions.
  • The three-dimensional set conjecture was resolved in a sequence of works of Hong Wang and Joshua Zahl, ending with the Hausdorff- and Minkowski-dimension theorem (arXiv:2502.17655); Guth, Wang and Zahl gave a streamlined proof (arXiv:2601.14411). The three-dimensional manuscript notes that the maximal conjecture requires a union estimate with the cube of the shading density, whereas the set theorem gives a weaker power, and that Wang and Zahl state the need for this replacement after their theorem.
  • In four dimensions, Wolff's bound gives dimension three; Łaba and Tao improved the upper Minkowski dimension. The polynomial Wolff estimates of Guth and Zahl with work of Zahl and of Katz and Rogers gave 3 + 1/40 for compact sets, Katz and Zahl's planebrush gave Hausdorff dimension at least 3.059, and Borges, Chan, Chen, Liu, Xi and Zhan improved the maximal parameter to about 3.0543. Rai Choudhuri proved the bound 13/4 for compact sticky sets.

The original results

• The Kakeya maximal conjecture in three dimensions (23 September 2026). Theorem 1.1 of the manuscript states that for every ε > 0 there is a constant C_ε such that the maximal average over unit tubes of radius δ in R³ satisfies ‖K_δf‖_L³(S²) ≤ C_εδ^−ε‖f‖_L³(R³) for all f and 0 < δ < 1.
• Kakeya sets in four dimensions (24 September 2026). Theorem 1.1 of the second manuscript states that every subset of R⁴ containing a unit line segment in every direction has Hausdorff dimension four, with no compactness, measurability or regularity assumption on the set or on its witnessing segments. A final section derives consequences, including packing and bounded Minkowski dimension statements, a lower bound in higher dimensions by projection, and Nikodym and curved-Kakeya applications through cited transfer results.
  • Method, as described: the three-dimensional proof combines the set estimate of Guth, Wang and Zahl, which applies under a convex-clustering bound, with multilinear Kakeya and the planar Furstenberg theorem, and keeps the cubic dependence on the density of the shadings through the passage to arbitrary L³ functions. The four-dimensional proof uses a weighted plank estimate from the three-dimensional manuscript (its Lemma 2.3) as its three-dimensional input.
• Sources. Commit 1581d4c of the openai/math repository, folders The Kakeya maximal conjecture in three dimensions and Every four-dimensional Kakeya set has full Hausdorff dimension.
• Formalization. The collection has no Lean scope note in lean/docs/ for this family.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 18 October 2026)
2026-10-16	J. Jardine, Lattice24, referee reviews rank_213 (three dimensions) and rank_028 (four dimensions) in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	Both reviews report no gap. The review of the three-dimensional manuscript describes partial coverage of a manuscript of about 7,700 lines. The review of the four-dimensional manuscript notes that its reduction rests on the weighted plank estimate cited from the three-dimensional manuscript, which that review did not audit.	Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.

Notes

• Related public work that does not refer to the manuscripts: M. Rai Choudhuri, arXiv:2610.12480 (submitted 30 August 2026), gives the Hausdorff-dimension bound 3.073 for Kakeya sets in R⁴; T. Moreira, arXiv:2610.06181 (5 October 2026), extends the Kakeya maximal estimate in R⁴ to dimension 3.06737 together with an improved restriction range.
• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.