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The three-dimensional Kakeya conjecture, after Wang and Zahl

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The Wang–Zahl proof of the three-dimensional Kakeya conjecture can be adapted to prove the conjecture in higher dimensions.
Without evidence. Tao gives a reasoned opinion rather than a result: the induction on scales framework should carry over, but degenerate cases that reduced to planar L^2 arguments in three dimensions would in four dimensions require the full three-dimensional machinery, and the right induction hypothesis is not yet known. He flags possible unfavourable surprises in the exponent numerology and, in a later reply, that quadric hypersurfaces are likely to appear in higher dimensions. The comment thread also contains a reply, apparently from Zahl, noting that higher-dimensional grains need not be pieces of hyperplanes. Sep 15, 2026

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Hong Wang and Joshua Zahl have just released a preprint that resolves the three-dimensional case of the infamous Kakeya set con…

The three-dimensional Kakeya conjecture has been proved: every three-dimensional Kakeya set has dimension three.Verified

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In principle if one had exactly the right induction hypothesis in the ambient dimension, one could then get Kakeya estimates in all dimensions, but finding precisely the right hypothesis for which the induction will close might take some time to work out. But I think this will get done eventually; the obstructions I see to extending to higher dimensions are primarily technical in nature, rather than fundamental obstacles (though there could be some nasty unfavorable surprises in the exponent numerology when one tries to execute the strategy carefully).

The Wang–Zahl proof of the three-dimensional Kakeya conjecture can be adapted to prove the conjecture in higher dimensions.SupportedReading · this claim’s Steward

Without evidence. Tao gives a reasoned opinion rather than a result: the induction on scales framework should carry over, but degenerate cases that reduced to planar L^2 arguments in three dimensions would in four dimensions require the full three-dimensional machinery, and the right induction hypothesis is not yet known. He flags possible unfavourable surprises in the exponent numerology and, in a later reply, that quadric hypersurfaces are likely to appear in higher dimensions. The comment thread also contains a reply, apparently from Zahl, noting that higher-dimensional grains need not be pieces of hyperplanes.

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Link: https://terrytao.wordpress.com/2025/02/25/the-three-dimensional-kakeya-conjecture-after-wang-and-zahl/
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Sep 15, 2026ReadingSteward of “The Wang–Zahl proof of the three-dimensional Ka…” · asserts without evidence, read whole
Sep 15, 2026Entered the graphEntered the graph; copy stored