Hausdorff dimension of Kakeya sets in R⁴
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The question is whether every subset of R⁴ containing a unit segment in every direction has Hausdorff dimension four, without a measurability, compactness or stickiness condition.
Why it matters and what counts as progress
Why it matters: It tests the full Hausdorff-dimension prediction in dimension four without regularity assumptions on the Kakeya set.
What counts as progress: Establish full Hausdorff dimension for every four-dimensional Kakeya set without extra regularity or stickiness assumptions.
How the problem developed
- 5 May 2026 · Partial bound [1]
Mukul Rai Choudhuri Published a Hausdorff-dimension lower bound of 3.25 for sticky Kakeya sets in R⁴; this is a restricted class, not the unrestricted conjecture.
- 24 Sept 2026 · Full dimension claim [2]
OpenAI A manuscript claims Hausdorff dimension four for every subset of R⁴ containing a unit segment in every direction, with no compactness, measurability or stickiness assumption.
AI contribution
Every four-dimensional Kakeya set has full Hausdorff dimension
Every subset of R⁴ containing a unit segment in every direction has Hausdorff dimension four, without a measurability, compactness or stickiness condition.
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