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Math-2506 · mathematics · Analysis

Higher-dimensional Kakeya conjecture

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For every dimension $n\geq4$, must every subset of $\mathbb R^n$ containing a unit segment in each direction have full dimension $n$ in the Kakeya conjecture? The question covers all higher dimensions, not only one fixed dimension.

Formulation references: [S1], [S2].

Short English statement

For every dimension $n\geq4$, must every subset of $\mathbb R^n$ containing a unit segment in each direction have full dimension $n$ in the Kakeya conjecture? The question covers all higher dimensions, not only one fixed dimension.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2506-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2506

Four-dimensional Kakeya sets: The manuscript’s result proves the Hausdorff dimension claim for n=4; the question higher-dimensional assertion also quantifies over all n>=5.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Higher-dimensional Kakeya conjecture

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

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