Kakeya sets in ℝ⁴ have Hausdorff dimension 4, and the Kakeya maximal function in ℝ⁴
Same grades as the calibrated Kakeya entry (62/10/53), which the calibration pass set from the accepted 3D proof and the 4D sticky case rather than the unrefereed full claim. The 4D accepted bound (3.073) is far from 4 and the non-sticky reduction is missing, but the tools are fresh, so it sits below restriction in ℝ³ (72) and just above the zero-density problem (61). Leverage 53 reflects restriction, Bochner–Riesz and maximal estimates that depend on Kakeya-type bounds.
The problem
A needle that points in every direction in 4D space must sweep out a set of full 4D size. Proved only in 2D and 3D.
- 1A needle must point in every direction
- 2Area can be tiny, so use dimension
- 3Conjecture: dimension is the full n
- 4Proved in 2D and 3D; 4D is open
Statement. A Kakeya set contains a unit line segment in every direction. The says every such has Hausdorff and Minkowski dimension . The stronger asserts (here for , the exponent is ), where is the supremum, over -thick tubes of direction , of the average of over the tube. In words: a function cannot have large averages along tubes pointing in all directions unless it is large overall.
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Step 2
- Harmonic analysisTheory · host
- Then
- Kakeya sets in ℝ⁴ have Hausdorff dimension 4, and the Kakeya maximal function in ℝ⁴The problem
State of the art
Open in the accepted literature for n = 4; a full claim is unverified. Davies (1971) proved n = 2. Wang and Zahl (arXiv:2502.17655, 2025) proved that every Kakeya set in ℝ³ has Minkowski and Hausdorff dimension 3, reducing to the sticky case (arXiv:2210.09581, J. Amer. Math. Soc. 39 (2026)); the streamlined proof of Guth, Wang and Zahl (arXiv:2601.14411) is to appear in Mathematika, the IMU's 2026 Fields Medal citation for Hong Wang includes it, and a Lean formalization (Project Numina with Nankai University and ByteDance Seed Prover) was released in September 2026. In ℝ⁴, Wolff's hairbrush gives (n+2)/2 = 3; Katz and Zahl (arXiv:1902.00989, Rev. Mat. Iberoam. 2021) improved this to about 3.059 with planebrushes, and Choudhuri (arXiv:2610.12480, v1 30 August 2026, unrefereed) obtains 3.073 by feeding the three-dimensional theorem into the planebrush argument. Wang and Zakharov (arXiv:2609.22035, September 2026) prove the four-dimensional conjecture for sticky families of tubes satisfying convex Wolff axioms. OpenAI's 6 October 2026 release (github.com/openai/math, family 074) includes manuscripts claiming the Hausdorff-dimension conjecture in ℝ⁴ and the Kakeya maximal conjecture in ℝ³; no Lean formalization is listed for them, they are unrefereed, and no independent assessment had appeared by mid-October 2026. Dvir (2008) proved the finite-field analogue, and the Heisenberg-group example of Katz, Łaba and Tao shows that a proof must use real geometry.
Realistic applications
Kakeya estimates control Fourier restriction, Bochner–Riesz summability and Bourgain's line of results on dispersive PDEs; resolution in all dimensions would settle the endpoint cases of the restriction hierarchy, but there is no engineering application.
What counts as solved
A refereed or independently verified proof that every Kakeya set in ℝ⁴ has Hausdorff dimension 4 (or the maximal function bound with ε-loss). Results for sticky or special families, or claims with no independent assessment, do not resolve it.
How it gets verified
Lean proofA formal proof checked by Lean 4 / Mathlib.
Statement: Kakeya.kakeya_set_conjecture 4, i.e. KakeyaSetConjectureDim 4, in Wikipedia/Kakeya.lean (Formal Conjectures pinned at commit e754a78 (17 October 2026)): every Kakeya set in ℝ⁴ has Hausdorff dimension 4, using Mathlib's dimH; the cases n = 2 and n = 3 are labelled solved in the same file. A proof is accepted when it compiles against the pinned corpus and passes Comparator. The pack's sharper maximal-function form and the Minkowski-dimension version are not stated in the corpus, so progress on those alone is refereed.
Claims under verification
+ add a claimNo claimed results on the Verification board for this problem yet.
Where the work is
This is geometric measure theory and incidence geometry, with formalization as a minor future share.
Bottlenecks
ranked by votes- #1TheoryHigher-dimensional Wolff axioms and non-sticky reduction
The n = 3 proof reduces the general case to sticky Kakeya via convex Wolff axioms. In ℝ⁴ the Wolff-type structure of tube families has more degrees of freedom, and the reduction is not available.
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Connected problems
- This problem is related to Stein restriction conjecture for the paraboloid in ℝ³: the extension operator maps L^∞ to L^p for all p > 3
Progress
- Oct 6, 2026Unrefereed OpenAI claim in ℝ⁴source
Full Hausdorff-dimension conjecture claimed in a model-produced manuscript; no independent assessment yet.
- Sep 18, 2026Sticky case in ℝ⁴source
Wang–Zakharov prove the conjecture for sticky tube families with convex Wolff axioms.
- Feb 24, 2025Kakeya sets in ℝ³ have dimension 3source
Wang–Zahl, 127 pages.
- Feb 3, 2019Planebrush bound in ℝ⁴source
Katz–Zahl improve Wolff’s dimension 3 to about 3.059 (arXiv v1 February 2019).