{"id":"openai-math/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension","pdf":"https://github.com/openai/math/blob/main/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026","tags":["harmonic-analysis","kakeya-problem","fractal-geometry"],"score":0,"title":"Every four-dimensional Kakeya set has full Hausdorff dimension","domain":"Real and complex analysis","family":"074","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CA","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-24","reactions":{},"discussion":[{"url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/#:~:text=The%20release%20announces%20the%20maximal%20conjecture%20in%20three%20dimensions","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"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…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans"}],"formalized":false,"family_title":"Kakeya in three and four dimensions"}