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.

Manuscripts (2)

The Kakeya maximal conjecture in three dimensions

September 23, 2026 · source · cite as \cite{OAI:The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026}

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}.

Every four-dimensional Kakeya set has full Hausdorff dimension

September 24, 2026 · source · cite as \cite{OAI:Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026}

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.

Result 074 · Real and complex analysis. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.