Submit progress

on Kakeya needle problem (AlphaEvolve problem 9)

The statement

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.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.