[#P30] Kakeya conjecture in dimensions at least four
Problem. Let \(d\ge4\). Every set \(E\subset\mathbb{R}^d\) that contains a unit line segment in every direction has Hausdorff dimension \(d\).
1Context
Kakeya estimates connect geometric measure theory with oscillatory integrals, restriction theory, and partial differential equations.
2Problem setup
Definition 1 (A Kakeya set contains a unit line segment parallel to every possible direction). A Kakeya set contains a unit line segment parallel to every possible direction.
Definition 2 (Hausdorff dimension extends ordinary dimension to highly irregular sets). Hausdorff dimension extends ordinary dimension to highly irregular sets.
Remark 1. Kakeya estimates connect geometric measure theory with oscillatory integrals, restriction theory, and partial differential equations.
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
Current 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]
1Records
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
- Bochner-Riesz conjecture in higher dimensionsharmonic analysis
- Openness of convolution on l1 of the integersharmonic analysis
- Sharp fourth-power norm of the cyclic Hilbert transform at order 31harmonic analysis
How to cite
TheoremDB contributors, “Kakeya conjecture in dimensions at least four,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/kakeya-conjecture-in-dimensions-at-least-fourThis page as plain text: kakeya-conjecture-in-dimensions-at-least-four.md
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.
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