OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
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_ε(δ^{-ε}) for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.
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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_εδ^{-ε}.
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We prove the four-dimensional Hausdorff-dimension Kakeya conjecture: every subset of ℝ^(4) 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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