Math-2948 · mathematics · Analysis

Hausdorff dimension of four-dimensional Kakeya sets

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.

Must every subset E of $R^4$ that contains a unit line segment in every direction have Hausdorff dimension four?

Short English statement

Must every subset E of $R^4$ that contains a unit line segment in every direction have Hausdorff dimension four?

Sources

S1 Original question source or mathematical context. https://www.math.ucla.edu/~tao/preprints/Expository/kakeya.pdf.
S2 Original question source or mathematical context. https://doi.org/10.4171/owr/2026/4.
S3 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/README.md.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2948-A1 · Literature report · LLM-generated · unknown · 2026-10-06

OpenAI claim summary for Math-2948

Four-dimensional Kakeya sets: The direction family consists of all directions in four-space, and no regularity or compactness is imposed on the containing set. The conclusion is the exact Hausdorff dimension four, without an additional measurable-line-selection hypothesis.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Hausdorff dimension of four-dimensional Kakeya sets

OpenAI's claim

Four-dimensional Kakeya sets: The manuscript claims that a set in $\mathbb R^4$ containing a segment of length one in every direction has Hausdorff dimension four. It permits arbitrary sets and arbitrary choices of witnessing segments.

Mathematical statement of the claimed result

Mathematical claim 1.

If $K\subset\mathbb R^4$ and for each unit direction $e\in S^3$ some $a_e\in\mathbb R^4$ satisfies $\{a_e+te:0\le t\le1\}\subset K$, then $\dim_H K=4$. No measurability assumption on $K$ or regularity assumption on the choice $e\mapsto a_e$ is imposed. This theorem treats dimension four, rather than all higher dimensions.

Outline of the argument

Four-dimensional Kakeya sets: A hypothetical dimension loss is converted into a weighted incidence model for lines. Nested boxes and quadratic fits describe persistent concentration; each fit must retain enough incidence mass to contribute. Scalar comparison and planar rigidity then contradict the simultaneous density and time-length requirements forced on substantial terminal chart systems.

Formal verification

Four-dimensional Kakeya sets: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

OpenAI, Every Four-Dimensional Kakeya Set Has Full Hausdorff Dimension, 2026-09-24; citation key OAI:Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
OpenAI catalogue entry.

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Record details

Subjects: Analysis
Collection: Top Open Problems
Mathematics rank: 2948