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3D Kakeya problem

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open.

問題
Problem 10 in the AlphaEvolve repository: 3D Kakeya problem
貢献
AlphaEvolve (Google DeepMind) · LLM-guided evolutionary code search, with Georgiev, Gómez-Serrano, Tao and Wagner
検証
Construction can be checked directlyPreprint (arXiv, 2025)
経緯
Paper submitted 2025-11-03
出典
Problem page (AlphaEvolve repository)Paper (arXiv:2511.02864)
部分的
結果Improved bound · new construction

The problem

Let n≥2n \geq 2. Let C(n)C(n) denote the minimal volume ∣⋃j=1n⋃k=1nPj,k∣|\bigcup_{j=1}^n \bigcup_{k=1}^n P_{j,k}| of prisms Pj,kP_{j,k} with vertices

(xj,k,yj,k,0),(xj,k+1n,yj,k,0),(xj,k,yj,k+1n,0),(xj,k+1n,yj,k+1n,0), (xj,k+jn,yj,k+kn,1),(xj,k+j+1n,yj,k+kn,1),(xj,k+jn,yj,k+k+1n,0),(xj,k+j+1n,yj,k+k+1n,1)\begin{align*} &\quad (x_{j,k},y_{j,k},0), \left(x_{j,k}+\frac{1}{n},y_{j,k},0\right), \left(x_{j,k},y_{j,k}+\frac{1}{n},0\right), \left(x_{j,k}+\frac{1}{n},y_{j,k}+\frac{1}{n},0\right),\ &\left(x_{j,k}+\frac{j}{n},y_{j,k}+\frac{k}{n},1\right), \left(x_{j,k}+\frac{j+1}{n},y_{j,k}+\frac{k}{n},1\right), \left(x_{j,k}+\frac{j}{n},y_{j,k}+\frac{k+1}{n},0\right), \left(x_{j,k}+\frac{j+1}{n},y_{j,k}+\frac{k+1}{n},1\right) \end{align*}

for some real numbers xj,k,yj,kx_{j,k}, y_{j,k}. Establish upper and lower bounds for C(n)C(n) that are as strong as possible.

Statement quoted from problem 10 of the AlphaEvolve repository of problems (Google DeepMind, CC BY 4.0).

What AI did

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open. Status in the repository: New result. See Section 6.5 in "Mathematical Exploration and Discovery at Scale" paper. Records may have moved since; the repository's notes below list the latest bounds it knows of.

How it was checked

The results are explicit constructions or programs; the repository's notebooks include the verification code used to score them.

Limitations

An improved bound is progress, not a solution: the exact answer to the problem is still unknown. The authors report that results were better when experts set up and guided the search.

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