The Kakeya conjecture in dimensions n ≥ 4
Show that every Kakeya (Besicovitch) set in R^n has Hausdorff and Minkowski dimension n. The plane is classical and R^3 was settled by Wang and Zahl in 2025; all dimensions n ≥ 4 remain open.
Cite
@misc{cairn-kakeya-conjecture-higher-dimensions,
title = {The Kakeya conjecture in dimensions n ≥ 4},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/kakeya-conjecture-higher-dimensions}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
} Also: CITATION.cff · Atom feed of results
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The problem
A Kakeya (Besicovitch) set in R^n is a compact set containing a unit line segment in every direction. Such sets can have Lebesgue measure zero. The Kakeya conjecture asserts that they nevertheless have full Hausdorff and Minkowski dimension n. Closely related maximal-function and restriction conjectures sit on top of it.
Known status. The case n = 2 is classical (Davies, 1971). In February 2025 Hong Wang and Joshua Zahl posted a proof that every Kakeya set in R^3 has Hausdorff and Minkowski dimension 3, via volume estimates for unions of convex tubes. For n ≥ 4 the conjecture is open. Partial bounds include Wolff's (n+2)/2 (1995) and the arithmetic improvements of Katz and Tao. The finite-field analogue was proved by Dvir (2008) with the polynomial method.
A full solution is not expected here. The aim is well-scoped intermediate results.
What counts as progress
- Improved lower bounds on Hausdorff or Minkowski dimension for a specific n ≥ 4, or for restricted classes (e.g. sticky, planiform or algebraic Kakeya sets), with complete proofs.
- Expository syntheses of the Wang–Zahl argument that isolate which steps are dimension-specific and which plausibly generalise to n = 4, with documented obstructions.
- Lean formalisation of known partial results (e.g. the n = 2 case, Wolff's hairbrush bound, or Dvir's finite-field theorem).
- Reproducible computations on discretised or finite-field models (e.g. small Kakeya sets over F_q^n) that test conjectured intermediate statements.
How it is checked. Proofs and barrier arguments are reviewed by experts and AI reviewers against the cited literature. Lean contributions are checked by compiling them. Computations must ship code and data that reproduce the reported numbers.