Result 074, Real and complex analysis

Kakeya in three and four dimensions

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 L3(R3)L^3(\mathbb R^3) to L3(S2)L^3(S^2) with norm Oε(δ−ε)O_\varepsilon(\delta^{-\varepsilon}) for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.

Proof

The bigger picture

Why it matters

A Kakeya set contains a unit line segment in every direction, yet its size can be hard to pin down. These manuscripts claim sharp limits on such directional concentration in three and four dimensions.

What changes?

In three dimensions, maximal averages over unit-length, radius-delta tubes define an operator from L3 of space to L3 of the unit sphere. Its norm is at most C(epsilon) times delta to the power minus epsilon for every positive epsilon, with C(epsilon) depending only on epsilon. L3 measures size using cubed absolute values. In four dimensions, every Kakeya set is claimed to have Hausdorff dimension four, with no compactness or regularity assumptions on the set or its segment family.

What does that help mathematicians do?

Hausdorff dimension measures geometric size at arbitrarily fine scales. The four-dimensional claim rules out any lower-dimensional set containing segments in all directions, even when the segments are chosen without regularity. The three-dimensional estimate controls how strongly tube averages can concentrate across directions as tubes narrow: the amplification grows more slowly than any fixed inverse power of the radius. These are distinct conclusions, not a four-dimensional maximal estimate.

Are there practical applications?

The immediate value is foundational. Researchers studying directional averages gain a quantitative bound on how spatial concentration affects averages across directions. Researchers studying irregular four-dimensional sets gain a dimension restriction without needing to justify extra regularity assumptions. These are mathematical tools for analysis and geometry, rather than demonstrated technological applications.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The Kakeya maximal conjecture in three dimensions

September 23, 2026 97 pages

We prove the Kakeya maximal conjecture in three dimensions. For every ε > 0, the maximal average over unit tubes of radius δ maps L3(R3)L^3(\mathbb R^3) to L3(S2)L^3(S^2) with norm at most Cεδ−εC_\varepsilon\delta^{-\varepsilon}.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Every four-dimensional Kakeya set has full Hausdorff dimension

September 24, 2026 175 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.