Every four-dimensional Kakeya set has full Hausdorff dimension
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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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
The release announces the maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four dimensions . Periodic tiling in dimension three. In An Aperiodic Monotile I also mentioned Greenfeld and Tao’s disproof of the periodic translational tiling conjecture. The new…