∑math20

Real and complex analysis · Result 074

Kakeya in three and four dimensions

2 papers

Resolves the Kakeya maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four. In three dimensions, the radius-δ tube maximal operator maps \(L^3(\mathbb R^3)\) to \(L^3(S^2)\) with norm \(O_\varepsilon(\delta^{-\varepsilon})\) for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 2

First page of The Kakeya maximal conjecture in three dimensions

The Kakeya maximal conjecture in three dimensions

OpenAI · September 23, 2026

We prove the Kakeya maximal conjecture in three dimensions. For every ε > 0, the maximal average over unit tubes of radius δ maps \(L^3(\mathbb R^3)\) to \(L^3(S^2)\) with norm at most \(C_\varepsilon\delta^{-\varepsilon}\).

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Cite
@misc{OAI:The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026,
  author = {{OpenAI}},
  title = {{The Kakeya maximal conjecture in three dimensions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf}{OAI:The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026}},
  year = {2026}
}
First page of Every four-dimensional Kakeya set has full Hausdorff dimension

Every four-dimensional Kakeya set has full Hausdorff dimension

OpenAI · September 24, 2026

We prove the four-dimensional Hausdorff-dimension Kakeya conjecture: every subset of ℝ4 containing a unit line segment in every direction has Hausdorff dimension four. No compactness or regularity assumption is imposed on the set or its witnessing line family.

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Cite
@misc{OAI:Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026,
  author = {{OpenAI}},
  title = {{Every Four-Dimensional Kakeya Set Has Full Hausdorff Dimension}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf}{OAI:Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.