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on Kakeya and Nikodym sets in finite fields (AlphaEvolve problem 1)

The statement

Let $d \geq 1$ , and let $q$ be a prime power. Let $\mathbb{F}_q$ be a finite field of order $q$ . A Kakeya set is a set $K$ that contains a line in every direction, and an Nikodym set $N$ is a set with the property that every point $x$ in $\mathbb{F}_q^d$ is contained in a line that is contained in $N \cup \{x\}$ . Let $C^K(d,q), C^N(d,q)$ denote the least size of a Kakeya or Nikodym set in $\mathbb{F}_q^d$ respectively.

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.