Kakeya in three and four dimensions

What OpenAI claims

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) to L³(S²) with norm O_ε(δ^-ε) for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.

Papers (2)

1. The Kakeya maximal conjecture in three dimensions

   23 September 2026 · Preprint, not peer reviewed · PDF

   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) to L³(S²) with norm at most C_εδ^-ε.

2. Every four-dimensional Kakeya set has full Hausdorff dimension

   24 September 2026 · Preprint, not peer reviewed · PDF

   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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.