# Kakeya set conjecture, all dimensions

- kind: problem
- status: open
- year: 1917

**Statement.** Kakeya sets in $\mathbb{R}^n$ have Hausdorff dimension $n$.

Settled for n = 2 (Davies, 1971) and n = 3 (Wang and Zahl, 2025), and over finite fields (Dvir, 2008, with a startlingly short polynomial-method proof). The general Euclidean case is open. It sits under many problems in harmonic analysis, including restriction and Bochner–Riesz.

## What would settle it

Lean statement in formal-conjectures

## Sources

- [Wikipedia: Kakeya set](https://en.wikipedia.org/wiki/Kakeya_set)

[Geometry and topology](/wiki/Space:geometry-topology.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations to this page

- generalizes [kakeya-3d](/wiki/Problem:kakeya-3d.md) — The original curation records this as a special case of Kakeya set conjecture, all dimensions; the argument for the link has not been written yet.

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