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 $L^3(\mathbb R^3)$ to $L^3(S^2)$ 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.
released 2026-09-23 | 4 theorems · 34 lemmas · 49 proofs · 47,011 words |
PLAY LEVEL 1 »(pdf)
We prove the Kakeya maximal conjecture in three dimensions. For every ε > 0, the maximal average over unit tubes of radius δ maps $L^3(\mathbb R^3)$ to $L^3(S^2)$ with norm at most $C_\varepsilon\delta^{-\varepsilon}$.
released 2026-09-24 | 5 theorems · 41 lemmas · 71 proofs · 89,106 words |
PLAY LEVEL 2 »(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.