[#P30] Kakeya conjecture in dimensions at least four
Contents
Problem. Let \(d\ge4\). Every set \(E\subset\mathbb{R}^d\) that contains a unit line segment in every direction has Hausdorff dimension \(d\).
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Definitions and notation
1Context
Kakeya estimates connect geometric measure theory with oscillatory integrals, restriction theory, and partial differential equations.
2Remarks
Remark 1. A Kakeya set contains a unit line segment parallel to every possible direction.
Remark 2. Hausdorff dimension extends ordinary dimension to highly irregular sets.
3What counts as a solution
- Prove full Hausdorff dimension for every Kakeya set in every dimension n at least 4, or construct a Kakeya set in some such dimension with Hausdorff dimension below n.
1Status
Saved packet · July 31, 2026
Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Wang and Zahl prove the three-dimensional Kakeya conjecture. Guth's 2026 survey explains that proof and identifies dimension four as the next unresolved case; dimensions at least four remain open. Exact unresolved remainder: Prove full Hausdorff dimension for every Kakeya set in every dimension n at least 4, or construct a Kakeya set in one such dimension with dimension below n.[2][1]
1Packet records
Recent contributions
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Notes and companion material
Original intake status. The cited 2026 survey covers the proof in three dimensions; current sources report that dimensions n at least 4 remain open. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- The plane case is classical and the three-dimensional case was resolved by Wang and Zahl. This record begins at dimension four.
Computational notes
- Finite discretizations can test incidence estimates but do not determine the Hausdorff dimension of every Kakeya set.
2See also
- Flattest 32-term Littlewood polynomial on the unit circleharmonic analysis
- Sharp L2 norm of the centered maximal operator on C_31harmonic analysis
- Sharp fourth-power norm of the cyclic Hilbert transform at order 31harmonic analysis
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Cite this problem statement
Cite the original sources separately.
“Kakeya conjecture in dimensions at least four.” TheoremDB. P30. Problem statement; statement text SHA-256 8b84e1deaa2ef8ef411aaaf20ec8e9affab2c8d97d1d4eeb2c2c06200e37b0de. https://theoremdb.org/statement/?ref=P30
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title = {{Kakeya conjecture in dimensions at least four}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 8b84e1deaa2ef8ef411aaaf20ec8e9affab2c8d97d1d4eeb2c2c06200e37b0de},
url = {https://theoremdb.org/statement/?ref=P30}
}Plain text: Built Markdown snapshot
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
1References
- Packet source. Larry Guth, “The Kakeya conjecture, after Wang and Zahl”. arXiv:2604.03416 (2026). Larry Guth, arXiv:2604.03416, Bourbaki survey of the three-dimensional proof. ↗preprint · primary source · arXiv:2604.03416, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited 2026 survey covers the proof in three dimensions; current sources report that dimensions n at least 4 remain open. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Bourbaki survey of the three-dimensional theorem and discussion of higher dimensions.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Pins the new three-dimensional boundary and leaves dimensions at least four as the exact target.Source named by the research packet.
- Hong Wang and Joshua Zahl, “Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions”. arXiv:2502.17655 (2025). Theorem 1.1 on page 3. ↗preprint · primary source · arXiv source revision v1 · checked 2026-08-01Source use: original summary.Proves full Minkowski and Hausdorff dimension for Kakeya sets in R^3 and supplies no theorem for every dimension at least four.
An original CC0 restatement prepared by TheoremDB maintainers.
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