Submit progress

on 3D Kakeya problem (AlphaEvolve problem 10)

The statement

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

\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 $x_{j,k}, y_{j,k}$ . Establish upper and lower bounds for $C(n)$ 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.