Open problem, Analysis, Geometry, posed 1917

Kakeya conjecture

Partially solved

Every Besicovitch (or Kakeya) set $K \subseteq \mathbb{R}^n$ —that is, a Borel set containing a unit line segment in every direction $e \in S^{n-1}$ —has Hausdorff dimension and Minkowski dimension equal to $n$ : $\dim_H(K) = \dim_M(K) = n$ .

Research frontier as of 2026

As of 2026, the Kakeya conjecture is proved for $n = 2$ (Davies, 1971) and resolved in a landmark 127-page February 2025 preprint by Hong Wang and Joshua Zahl (`arXiv:2502.17655`) for $n = 3$ , proving $\dim_H(K) = \dim_M(K) = 3$ for every Kakeya set in $\mathbb{R}^3$ . Wang and Zahl combined their earlier proof of the sticky Kakeya conjecture (`arXiv:2210.09581`) with multiscale volume estimates for unions of $\delta$ -tubes packed inside convex sets. For all dimensions $n \ge 4$ , both the Hausdorff and Minkowski Kakeya conjectures remain open, as does the stronger Kakeya maximal function conjecture in $L^3(\mathbb{R}^3)$ and higher.

Best known results

In $\mathbb{R}^3$ , every Kakeya set has Hausdorff and Minkowski dimension $3$ (Hong Wang and Joshua Zahl, 2025 preprint `arXiv:2502.17655`).
In $\mathbb{R}^4$ , planebrush and polynomial-partitioning methods give $\dim_H(K) \ge 3.059$ (Katz and Zahl, 2021).
In general dimension $\mathbb{R}^n$ , arithmetic Kakeya reductions of Katz and Tao (2002) give $\dim_H(K) \ge (2 - \sqrt{2})(n - 4) + 3$ .

Tools and where they stop

Tool	Achieved	Where it stops
Multiscale tube-packing and sticky Kakeya analysis	Rules out hypothetical lower-dimensional configurations of $\delta$ -tubes in $\mathbb{R}^3$ by reducing general tube packings to sticky/convex-grain cases and establishing sharp volume bounds (Wang–Zahl, 2025).	In $\mathbb{R}^n$ for $n \ge 4$ , tubes can concentrate inside intermediate-dimensional algebraic subvarieties of dimension $2, \dots, n-2$ , creating much more complex incidence hierarchies.
Polynomial method and decoupling (Dvir, Guth, Bourgain–Demeter)	Completely solves the finite-field Kakeya problem in $\mathbb{F}_q^n$ (Dvir, 2009) and proves sharp $\ell^2$ decoupling inequalities and multilinear Kakeya estimates in $\mathbb{R}^n$ .	In Euclidean space, $\delta$ -tubes have thickness $\delta > 0$ and can intersect at small angles in semi-algebraic neighborhoods where polynomial vanishing arguments lose degree efficiency.

Open questions

Does every Kakeya set in $\mathbb{R}^n$ for $n \ge 4$ have Hausdorff and Minkowski dimension equal to $n$ ?
Can the Wang–Zahl three-dimensional Kakeya theorem be upgraded to the full $L^3(\mathbb{R}^3)$ Kakeya maximal function bound and the full three-dimensional Fourier restriction conjecture?

Proofs

References

Hong Wang, Joshua Zahl (2025). Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions · arXiv:2502.17655 [preprint, not peer-reviewed]
Thomas Wolff (1999). An algorithmic approach to the Kakeya problem · DOI:10.1090/ulect/029/10
Zeev Dvir (2009). On the size of Kakeya sets in finite fields · DOI:10.1090/S0894-0347-08-00607-3