# P30: Kakeya conjecture in dimensions at least four

- ID: `P30`
- Reference: `kakeya-conjecture-in-dimensions-at-least-four`
- Page: https://theoremdb.org/statements/P30
- Record maturity: Reviewed problem with recorded work

## Problem

Let \(d\ge4\). Every set \(E\subset\mathbb{R}^d\) that contains a unit line segment in every direction has Hausdorff dimension \(d\).

### Context

Kakeya estimates connect geometric measure theory with oscillatory integrals, restriction theory, and partial differential equations.

### Problem setup

- **Definition (A Kakeya set contains a unit line segment parallel to every possible direction).** A Kakeya set contains a unit line segment parallel to every possible direction.
- **Definition (Hausdorff dimension extends ordinary dimension to highly irregular sets).** Hausdorff dimension extends ordinary dimension to highly irregular sets.
- **Remark.** Kakeya estimates connect geometric measure theory with oscillatory integrals, restriction theory, and partial differential equations.

### What counts as a solution

- Prove full Hausdorff dimension for every Kakeya set in every dimension n at least 4, or construct a Kakeya set in some such dimension with Hausdorff dimension below n.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: Wang and Zahl prove the three-dimensional Kakeya conjecture. Guth's 2026 survey explains that proof and identifies dimension four as the next unresolved case; dimensions at least four remain open. Exact unresolved remainder: Prove full Hausdorff dimension for every Kakeya set in every dimension n at least 4, or construct a Kakeya set in one such dimension with dimension below n. [2](#reference-2) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: Wang and Zahl prove the three-dimensional Kakeya conjecture. Guth's 2026 survey explains that proof and identifies dimension four as the next unresolved case; dimensions at least four remain open. Exact unresolved remainder: Prove full Hausdorff dimension for every Kakeya set in every dimension n at least 4, or construct a Kakeya set in one such dimension with dimension below n.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: Wang and Zahl prove the three-dimensional Kakeya conjecture. Guth's 2026 survey explains that proof and identifies dimension four as the next unresolved case; dimensions at least four remain open.

Exact unresolved remainder: Prove full Hausdorff dimension for every Kakeya set in every dimension n at least 4, or construct a Kakeya set in one such dimension with dimension below n.

### Background and intake notes

- Original intake status: The cited 2026 survey covers the proof in three dimensions; current sources report that dimensions n at least 4 remain open. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The dimensional scope was checked against the cited survey and current literature on 2026-07-22.
- The plane case is classical and the three-dimensional case was resolved by Wang and Zahl. This record begins at dimension four.

### Open directions

- **Route 1** (reported): Prove full Hausdorff dimension for every Kakeya set in every dimension n at least 4, or construct a Kakeya set in some such dimension with Hausdorff dimension below n. [1](#reference-1)

### Computational notes

- Finite discretizations can test incidence estimates but do not determine the Hausdorff dimension of every Kakeya set.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `kakeya-conjecture-in-dimensions-at-least-four`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Larry Guth, “The Kakeya conjecture, after Wang and Zahl”. arXiv:2604.03416 (2026). Larry Guth, arXiv:2604.03416, Bourbaki survey of the three-dimensional proof https://arxiv.org/abs/2604.03416
   - Also cited at Bourbaki survey of the three-dimensional theorem and discussion of higher dimensions
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2604.03416, checked 2026-07-31; checked 2026-07-31
   - Source use: original_summary
   - The cited 2026 survey covers the proof in three dimensions; current sources report that dimensions n at least 4 remain open. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - Pins the new three-dimensional boundary and leaves dimensions at least four as the exact target.
   - Source named by the research packet.
2. <a id="reference-2"></a>Hong Wang and Joshua Zahl, “Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions”. arXiv:2502.17655 (2025). Theorem 1.1 on page 3 https://arxiv.org/abs/2502.17655
   - preprint; primary source; arXiv source revision v1; checked 2026-08-01
   - Source use: original_summary
   - Proves full Minkowski and Hausdorff dimension for Kakeya sets in R^3 and supplies no theorem for every dimension at least four.