Kakeya and Nikodym sets in finite fields

The problem

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.

Statement quoted from problem 1 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 6.1 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.

Notes from the repository

This work led to the spinoff paper "New Nikodym set constructions over finite fields"; see this blog post for more details.

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.

∎