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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

L^3(\mathbb R^3)

L^3(S^2)

with norm

O_\varepsilon(\delta^{-\varepsilon})

for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.

Manuscripts in this family · 2

The Kakeya maximal conjecture in three dimensions

Sep 23, 2026·paper.pdf

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)

L^3(S^2)

with norm at most

C_\varepsilon\delta^{-\varepsilon}

Full abstract

PDF on GitHub LaTeX source

Every four-dimensional Kakeya set has full Hausdorff dimension

Sep 24, 2026·paper.pdf

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.

Full abstract

PDF on GitHub LaTeX source