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Kakeya Conjecture — Does a Needle-Set Have Full Dimension?
Problem
A Kakeya set in contains a unit segment in every direction. Besicovitch (1928) showed such sets can have Lebesgue measure zero. The Kakeya conjecture: every Kakeya set has Minkowski and Hausdorff dimension exactly .
Why it matters
Davies proved the planar Hausdorff-dimension case in 1971. A Wang–Zahl preprint dated 2025-02-24 gave a proof in three dimensions, followed by detailed expository verification and surveys. The case remains open. Kakeya estimates are deeply connected with restriction and other harmonic-analysis problems, but those families of formulations are not blanket equivalents.
Progress so far
was solved by Davies (1971). Wang and Zahl presented a proof for in a 2025 arXiv preprint; Guth's proof outline and later surveys provide follow-up exposition. We retain that publication boundary rather than presenting the preprint as a cited journal article. For the conjecture remains open.
Further reading
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