Kakeya needle problem

The problem

Let $n \geq 2$. Let $C^T(n)$ denote the minimal area $|\bigcup_{j=1}^n T_j|$ of a union of triangles $T_j$ with vertices $(x_j,0)$, $(x_j + 1/n, 0)$, $(x_j + j/n, 1)$ for some real numbers $x_1,\dots,x_n$, and similarly define $C^P(n)$ denote the minimal area $|\bigcup_{j=1}^n P_j|$ of a union of parallelograms $P_j$ with vertices $(x_j,0), (x_j+1/n,0), (x_j+j/n,1), (x_j+(j+1)/n,0)$ for some real numbers $x_1,\dots,x_n$. Finally, define $S^T(n)$ to be the maximal "score"

$$\frac{\sum_{i=1}^n |T_i|}{\left(\sum_{i=1}^n \sum_{j=1}^n |T_i \cap T_j|\right)^{1/2} |\bigcup_{i=1}^n T_i|^{1/2}}$$

over triangles $T_i$ as above, and define $S^P(n)$ similarly. Establish upper and lower bounds for $C^T(n)$, $C^P(n)$, $S^T(n)$, $S^P(n)$ that are as strong as possible.

Statement quoted from problem 9 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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