Kakeya set conjecture in Euclidean space

Establish that every Besicovitch set K ⊂ R^n has Minkowski and Hausdorff dimension equal to n, thereby resolving the Kakeya set conjecture in all dimensions.

Background

Besicovitch sets are compact subsets of Euclidean space that contain a unit line segment in every direction. In R2, Besicovitch constructed a null-measure example, motivating quantitative questions about compressibility of such sets at small scales.

The Kakeya set conjecture posits full dimension for all Besicovitch sets and is a cornerstone problem linked to Fourier analysis and geometric measure theory. It is proved in R2 (Davies; Córdoba) and, via Wang–Zahl, in R3, but remains open in higher dimensions.

References

Conjecture [Kakeya set conjecture] Every Besicovitch set $K\subsetRn$ has Minkowski and Hausdorff dimension $n$.

— A Survey of the Kakeya conjecture, 2000-2025  (2512.09397 - Zahl, 10 Dec 2025) in Conjecture \ref{kakeyaSetConj}, Section 1 (Introduction)

The Kakeya conjecture predicts that such a set must have full Hausdorff dimension. \begin{conjecture}[Kakeya conjecture] If $K\subset\mathbb{R}n$ is a Kakeya set, then $\dim_{\mathbb H}K=n$. \end{conjecture}

— Refinements on the Restriction and Kakeya problems range in $\mathbb{R}^4$  (2610.06181 - Moreira, 5 Oct 2026) in Section 1, introduction; Conjecture (Kakeya conjecture)