Kakeya Conjecture

The Kakeya conjecture (also called the Kakeya set conjecture or Besicovitch set conjecture) is a problem in geometric measure theory and harmonic analysis. It predicts that any set in $n$-dimensional Euclidean space containing a unit line segment pointing in every direction must have full dimension $n$, even though such sets can have zero volume.^[7][1] The question grew out of Sōichi Kakeya's 1917 problem about the smallest region in which a needle can be turned around.^[1][2] It is known to hold in two dimensions, through work of Roy Davies (1971), and in three dimensions, through a proof that Hong Wang and Joshua Zahl posted in February 2025.^[3][18][5] A stronger form, the Kakeya maximal function conjecture, sits at the base of a hierarchy of conjectures in Fourier analysis that also includes the restriction, Bochner-Riesz and local smoothing conjectures.^[6][7] On 26 October 2026 OpenAI released a manuscript, produced by an internal model, that claims to prove the local smoothing conjecture in every dimension and with it the Kakeya conjecture in all dimensions. As of 30 October 2026 that claim is an unrefereed preprint accompanied by a Lean formalization.^[23][24][26]

Statement

A Besicovitch set (or Kakeya set) is a subset of $\mathbb{R}^n$ that contains a unit line segment in every direction.^[7][10] A ball of diameter 1 is an example, but Besicovitch showed that such sets can have Lebesgue measure zero.^[4][7] The conjecture asks how small they can be when size is measured by dimension rather than volume.

Kakeya set conjecture. Every Besicovitch set $K \subset \mathbb{R}^n$ has Minkowski and Hausdorff dimension $n$.^[7]

The conjecture is usually studied at a small scale $\delta > 0$. The $\delta$-neighbourhood of a Besicovitch set contains a $\delta$-tube (a $\delta \times \cdots \times \delta \times 1$ cylinder around a unit segment) in each of about $\delta^{1-n}$ directions that are $\delta$-separated. The discretized conjecture says that the union of such a family of tubes must have volume at least $C_\varepsilon \delta^{\varepsilon}$ for every $\varepsilon > 0$, that is, almost as large as if the tubes did not overlap at all.^[7][8] Quanta Magazine describes the same idea as a configuration of overlapping tubes pointing everywhere, where the goal is to arrange them with as much overlap as possible.^[5]

Kakeya maximal function conjecture. For a locally integrable function $f$ on $\mathbb{R}^n$ and a direction $e$ on the unit sphere $S^{n-1}$, the Kakeya maximal function is

$K_\delta f(e) = \sup_{T \parallel e} \frac{1}{|T|} \int_T |f|,$

where the supremum runs over all $\delta$-tubes $T$ pointing in direction $e$. The conjecture states that for every $\varepsilon > 0$

$\|K_\delta f\|_{L^n(S^{n-1})} \le C_{n,\varepsilon}\, \delta^{-\varepsilon} \|f\|_{L^n(\mathbb{R}^n)}.$

The maximal function conjecture implies the Hausdorff dimension statement, which in turn implies the Minkowski dimension statement.^[7][10] Because of this ordering, a result about one form of the conjecture in a given dimension does not automatically settle the stronger forms. After Wang and Zahl's 2025 proof, for example, Terence Tao noted that the maximal function version in three dimensions remained open.^[8]

The Kakeya needle problem

In 1917 Kakeya asked for the smallest area in which a unit line segment (a "needle") can be rotated by 180 degrees in the plane. Rotating the needle about its midpoint sweeps a disk of area $\pi/4$, and a "three-point U-turn" needs only $\pi/8$.^[1] Kakeya's paper, "Some problems on maxima and minima regarding ovals", appeared in the Tohoku Science Reports.^[2] For convex regions the answer is the equilateral triangle of height 1, a result published by Gyula Pál in 1921.^[2]

Abram Besicovitch settled the general question. In a 1919 paper published in Perm, he constructed a set of measure zero containing a line segment in every direction while working on a problem about integration, and in a 1928 paper in Mathematische Zeitschrift he showed that a needle can be turned around inside a region of arbitrarily small area.^[3][4] Oskar Perron simplified the construction in 1928.^[3] Roy Davies described F. Cunningham Jr.'s later result, that the rotation can be done inside a simply connected subset of the unit circle with arbitrarily small measure, as "the last word on Kakeya's problem itself".^[3]

Connection to Fourier analysis

The question gained importance in analysis in 1971, when Charles Fefferman used Besicovitch's construction to prove that the ball multiplier (the operator that cuts off the Fourier transform of a function outside a ball) is unbounded on $L^p(\mathbb{R}^n)$ for every $p \neq 2$ when $n > 1$. The tubes of a Besicovitch-type arrangement can be chosen disjoint while their shifted copies overlap heavily, and this geometry produces the counterexample. One consequence is that Fourier inversion by summing over expanding balls does not converge in $L^p$ in higher dimensions unless $p = 2$.^[1][5]

Fefferman's theorem led to a family of conjectures that the Kakeya conjecture supports from below. Quanta Magazine describes the hierarchy as a tower: the restriction conjecture of Elias Stein implies the Kakeya conjecture, the Bochner-Riesz conjecture implies the restriction conjecture, and the local smoothing conjecture, formulated by Christopher Sogge, sits at the top.^[6] Tao's 2001 survey explains the mechanism behind two of these links: a counterexample to the Kakeya conjecture could be used to build counterexamples to the Bochner-Riesz and local smoothing conjectures.^[1] Because each statement implies the one below it, a disproof of Kakeya would have brought down the whole tower, while a proof of Kakeya does not by itself prove the conjectures above it.^[6]

In 1991 Jean Bourgain introduced a method that runs some of these implications in reverse, so that progress on the Kakeya problem yields progress on the Bochner-Riesz problem and related questions. Functions are decomposed into "wave packets" concentrated on tubes, and Kakeya-type estimates then control how much the packets can overlap.^[1][7]

Partial results

Before 2025 the conjecture was settled only in the plane. In higher dimensions, work concentrated on lower bounds for the dimension of Besicovitch sets and on partial maximal function estimates.

Zahl's 2025 survey of the field covers progress since the earlier surveys by Wolff and by Katz and Tao.^[7]

Finite field version

Wolff suggested studying Kakeya sets over finite fields: a subset of $\mathbb{F}_q^n$ that contains a line in every direction. He asked whether such a set must have at least $C_n q^n$ points, and the bound in his paper was of order $q^{(n+2)/2}$.^[12] In a preprint posted in March 2008, Dvir proved the bound $|K| \ge C_n q^n$, where $C_n$ depends only on $n$. His argument uses the polynomial method: a low-degree polynomial that vanishes on a Kakeya set must vanish identically, so the set cannot be smaller than the dimension of the space of such polynomials.^[12]

Three-dimensional proof

Wang (New York University's Courant Institute) and Zahl (University of British Columbia) posted "Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions" to arXiv on 24 February 2025. The abstract states that the union of a family of $\delta$-tubes in $\mathbb{R}^3$ has almost maximal volume when not too many tubes fit inside a common convex set, and that as a consequence every Kakeya set in $\mathbb{R}^3$ has Minkowski and Hausdorff dimension 3.^[18] Tao called it "spectacular progress in geometric measure theory" and noted that the proof is 127 pages long and relies on the authors' earlier proof of the sticky case.^[8]

The sticky case followed a program that Katz and Tao had laid out in 2014. For the remaining, non-sticky sets, Wang and Zahl worked with "grains", the small regions in which many tubes overlap that Guth had shown any counterexample must contain.^[5] The argument raises the bound step by step: starting from Wolff's bound of 5/2, it shows that any lower bound for the volume of a union of tubes can be improved by a small amount, and repeats this until the bound reaches 3.^[5][8] Nets Katz told Quanta that "it's a once-in-a-century kind of result".^[5] Zahl describes the proof as a trilogy of papers: the sticky case, the Assouad dimension result and the final paper.^[7][21] In January 2026 Larry Guth, Wang and Zahl posted "A streamlined proof of the Kakeya set conjecture in $\mathbb{R}^3$", revised on 11 October 2026.^[19]

The International Mathematical Union awarded Wang a 2026 Fields Medal. Its citation lists "the Kakeya problem in three dimensions" along with her work on local smoothing for the planar wave equation, Fourier restriction, Falconer distance sets and Furstenberg sets in the plane.^[20] The University of British Columbia stated that the prize was given in part for the Kakeya work with Zahl.^[21] Wang had earlier co-authored, with Guth and Ruixiang Zhang, the 2019 proof of the local smoothing conjecture in two spatial dimensions, published in the Annals of Mathematics in 2020.^[22]

AI-generated claims (2026)

September 2026 manuscripts

The OpenAI Mathematics Collection, released on 6 October 2026, contains manuscripts written by an internal OpenAI model, several of them on this problem. Result family 074, "Kakeya in three and four dimensions", groups "The Kakeya maximal conjecture in three dimensions" (dated 23 September 2026), which claims the $L^3$ maximal estimate with $\delta^{-\varepsilon}$ loss in $\mathbb{R}^3$, and "Every four-dimensional Kakeya set has full Hausdorff dimension" (dated 24 September 2026).^[27] The same initial release contained "Critical local smoothing for the three-dimensional wave equation", which claims Sogge's conjecture in three spatial dimensions, and a manuscript on Fourier restriction to the sphere in $\mathbb{R}^3$ for bounded data.^[27]

Local smoothing in every dimension

On 26 October 2026 OpenAI added "Local smoothing for the Euclidean wave equation" (dated 25 October 2026) to the repository.^[25] The manuscript claims Sogge's local smoothing conjecture for the Euclidean wave equation in every spatial dimension $n \ge 2$: for $2 \le p < \infty$ and every $\varepsilon > 0$, the half-wave evolution of the data, measured in $L^p(\mathbb{R}^n \times [1,2])$, is controlled by the data with $\max{0, \tfrac{n-1}{2} - \tfrac{n}{p}} + \varepsilon$ derivatives in $L^p$.^[24] From this it derives the Bochner-Riesz conjecture, sharp bounds for Stein's square function, Fourier restriction estimates for the sphere, compact paraboloid and truncated cone throughout the interior of the conjectured range, and the Kakeya and Nikodym maximal conjectures. In particular it concludes that every Euclidean Kakeya set has full Hausdorff dimension.^[24]

OpenAI's announcement said the local smoothing conjecture "implies the Kakeya conjecture and Stein's restriction conjecture", that its models "produced a proof of the local smoothing conjecture in every dimension and therefore also resolve the Kakeya conjecture and Stein's restriction conjecture in all dimensions", and that the proof "is based on a novel auxiliary partial differential equation". According to the company, it obtained the AI output on 18 October 2026, and the result was formalized in Lean the following day.^[23]

The manuscript does not reach Kakeya through tube geometry. It proves an $L^{p_c}$ estimate for waves at the critical exponent $p_c = 2n/(n-1)$ with a logarithmic loss in frequency, using a positive semidefinite, divergence-free tensor (a conserved current) built from the wave, together with a Dirichlet problem for a Monge-Ampère-type equation involving a quaternionic Hessian. Placing random signs on disjoint frequency caps then converts a spherical extension estimate into a bound on how much tubes can overlap.^[24] Its Kakeya maximal estimate has a logarithmic loss:

$\|K_\delta f\|_{L^n(S^{n-1})} \le C_n \left[\log(2/\delta)\right]^{4/p_c} \|f\|_{L^n(\mathbb{R}^n)}, \qquad 0 < \delta \le 1.$

This is stronger than the $\delta^{-\varepsilon}$ form of the conjecture. The manuscript adds that the logarithmic power must be at least $1 - 1/n$.^[24] In its discussion of earlier work, it credits Davies with full dimension in the plane and Wang and Zahl with the three-dimensional case.^[24]

Release and verification status

OpenAI said it consulted an expert about forming a collaboration around the result and, on their recommendation, released the AI output directly so that researchers could organize their own efforts to explain it. This differed from its handling of the Exponential Time Hypothesis result, which an external working group will examine. OpenAI says it does not seek academic credit for discoveries produced by its internal models, and it describes such manuscripts as "best understood as a form of raw research data".^[23]

The repository's Lean scope note for the local smoothing family states that the formalization covers the half-wave estimate in every dimension $n \ge 2$ for $2 \le p < \infty$, uniform Bochner-Riesz bounds, spherical extension estimates, Stein square-function bounds, the $L^n$ Kakeya maximal bound with $\delta^{-\varepsilon}$ loss, and the statement that every subset of $\mathbb{R}^n$ containing a unit segment in each direction has Hausdorff dimension $n$. It also lists a three-dimensional lower bound for the volume of $\delta^{-2}$ direction-separated tubes, of the form $c/(1+\log(1/\delta))^{66}$.^[26] What the repository's Comparator checks do and do not establish is discussed in the OpenAI Mathematics Collection article.

When it was released, the manuscript was an unrefereed preprint credited to OpenAI as author. News coverage on the day of the announcement reported the claim alongside OpenAI's Exponential Time Hypothesis and Yang-Mills results.^[28] For context on how the company's earlier releases were received, including corrections and withdrawals and the response of the Advisory Group on Mathematics and Artificial Intelligence, see the OpenAI Mathematics Collection article.

Continue reading

• LLocal Smoothing ConjectureThe local smoothing conjecture is a conjecture in harmonic analysis about solutions of the wave equation.Mentioned 13 times above
• OOpenAI Mathematics CollectionThe OpenAI Mathematics Collection is a public collection of mathematical manuscripts and proof artifacts produced by an internal OpenAI model. OpenAI announced the release on October 6, 2026.Mentioned 3 times above
• EExponential Time HypothesisThe exponential time hypothesis (ETH) is a conjecture in computational complexity theory about how fast the 3-SAT problem can be solved.Mentioned twice above
• Terence TaoTerence Tao (born 1975 in Adelaide, Australia) is a mathematician at the University of California, Los Angeles, where he is a Distinguished Professor and holds the James and Carol Collins Chair in the College…Mentioned twice above
• MMathlibMathlib (formally mathlib4, the active Lean 4 incarnation) is the community-maintained, open-source unified library of formalized mathematics for the lean theorem prover, and it is the largest single library…Listed in See also
• AAutomated Theorem ProvingAutomated theorem proving (ATP) is the use of computer programs to find or check mathematical proofs, expressing both the statement and the proof in a formal_logic so that correctness can be decided…Listed in See also

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• Mathematical reasoning in AI

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• OpenAI
• Lean (Theorem Prover)
• Advisory Group on Mathematics and Artificial Intelligence
• OpenAI Navier-Stokes Proposed Solution
• Yang-Mills Existence and Mass Gap

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References

1. ^1 ^2 ^3 ^4 ^5 ^6 ^7 ^8Terence Tao. "From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and PDE." arXiv:math/0008098, 2000; *Notices of the American Mathematical Society* 48 (2001). arxiv.org/...0008098
2. ^1 ^2 ^3 ^4 ^5Sergio Cabello, Otfried Cheong and Michael Gene Dobbins. "The Inverse Kakeya Problem." arXiv:1912.08477, 2019 (introduction and references to Kakeya 1917 and Pál 1921). arxiv.org/...1912.08477
3. ^1 ^2 ^3 ^4 ^5 ^6 ^7 ^8Roy O. Davies. "Some remarks on the Kakeya problem." *Proceedings of the Cambridge Philosophical Society* 69(3):417-421, May 1971. doi.org/...S0305004100046867
4. ^1 ^2 ^3 ^4Christopher J. Bishop. "Another Besicovitch-Kakeya set." Note, Stony Brook University (Theorem 1 and references to Besicovitch 1919 and 1928). math.stonybrook.edu/...kakeya.pdf
5. ^1 ^2 ^3 ^4 ^5 ^6 ^7Joseph Howlett. "'Once in a Century' Proof Settles Math's Kakeya Conjecture." *Quanta Magazine*, 14 March 2025. quantamagazine.org/...s-kakeya-conjecture-20250314
6. ^1 ^2 ^3Jordana Cepelewicz. "A Tower of Conjectures That Rests Upon a Needle." *Quanta Magazine*, 12 September 2023. quantamagazine.org/...rests-upon-a-needle-20230912
7. ^1 ^2 ^3 ^4 ^5 ^6 ^7 ^8 ^9 ^10 ^11Joshua Zahl. "A Survey of the Kakeya conjecture, 2000-2025." arXiv:2512.09397, December 2025. arxiv.org/...2512.09397
8. ^1 ^2 ^3 ^4 ^5 ^6Terence Tao. "The three-dimensional Kakeya conjecture, after Wang and Zahl." *What's new* (blog), 25 February 2025. terrytao.wordpress.com/...ture-after-wang-and-zahl
9. ^Thomas Wolff. "An improved bound for Kakeya type maximal functions." *Revista Matemática Iberoamericana* 11(3):651-674, 1995. doi.org/...188
10. ^1 ^2 ^3 ^4 ^5 ^6 ^7Nets Hawk Katz and Terence Tao. "New bounds for Kakeya problems." arXiv:math/0102135, February 2001. arxiv.org/...0102135
11. ^Nets Hawk Katz, Izabella Łaba and Terence Tao. "An improved bound on the Minkowski dimension of Besicovitch sets in $\mathbb{R}^3$." arXiv:math/9903166, 1999; *Annals of Mathematics* 152(2):383-446, 2000. arxiv.org/...9903166
12. ^1 ^2 ^3Zeev Dvir. "On the size of Kakeya sets in finite fields." arXiv:0803.2336, March 2008. arxiv.org/...0803.2336
13. ^Nets Hawk Katz and Joshua Zahl. "An improved bound on the Hausdorff dimension of Besicovitch sets in $\mathbb{R}^3$." arXiv:1704.07210, April 2017. arxiv.org/...1704.07210
14. ^Larry Guth. "Degree reduction and graininess for Kakeya-type sets in $\mathbb{R}^3$." arXiv:1402.0518, February 2014. arxiv.org/...1402.0518
15. ^Jonathan Hickman, Keith M. Rogers and Ruixiang Zhang. "Improved bounds for the Kakeya maximal conjecture in higher dimensions." arXiv:1908.05589, August 2019. arxiv.org/...1908.05589
16. ^Hong Wang and Joshua Zahl. "Sticky Kakeya sets and the sticky Kakeya conjecture." arXiv:2210.09581, October 2022. arxiv.org/...2210.09581
17. ^Hong Wang and Joshua Zahl. "The Assouad dimension of Kakeya sets in $\mathbb{R}^3$." arXiv:2401.12337, January 2024; *Inventiones Mathematicae* 241(1):153-206, 2025. arxiv.org/...2401.12337
18. ^1 ^2 ^3Hong Wang and Joshua Zahl. "Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions." arXiv:2502.17655, 24 February 2025. arxiv.org/...2502.17655
19. ^Larry Guth, Hong Wang and Joshua Zahl. "A streamlined proof of the Kakeya set conjecture in $\mathbb{R}^3$." arXiv:2601.14411, January 2026 (version 2, 11 October 2026). arxiv.org/...2601.14411
20. ^International Mathematical Union. "Fields Medals 2026." mathunion.org/...fields-medals-2026
21. ^1 ^2 ^3 ^4Department of Mathematics, University of British Columbia. "The Kakeya Conjecture: A Mathematical Problem with a Long History at UBC Math and a 2026 Fields Medal Connection." math.ubc.ca/...4209
22. ^Larry Guth, Hong Wang and Ruixiang Zhang. "A sharp square function estimate for the cone in $\mathbb{R}^3$." *Annals of Mathematics* 192(2):551-581, 2020; arXiv:1909.10693. annals.math.princeton.edu/...p06
23. ^1 ^2 ^3OpenAI. "Working with academic communities to build scientific understanding." 26 October 2026. openai.com/...erstanding-with-academic-communities
24. ^1 ^2 ^3 ^4 ^5 ^6 ^7OpenAI. "Local smoothing for the Euclidean wave equation." Manuscript dated 25 October 2026, openai/math repository. github.com/...paper.pdf
25. ^OpenAI. openai/math repository history (entry for 26 October 2026). github.com/...history.md
26. ^1 ^2OpenAI. "Local smoothing for the Euclidean wave equation" (Lean formalization scope, document 079). github.com/...079.md
27. ^1 ^2OpenAI. openai/math manuscript map (families 074 and 079). Accessed 30 October 2026. github.com/...CONTENTS.md
28. ^"OpenAI says its AI model has disproved the Exponential Time Hypothesis." *The Next Web*, 26 October 2026. thenextweb.com/...pothesis-false-kakeya-yang-mills

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