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OP-12106Status unassessedCombinatorics

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

Kakeya and Nikodym sets in finite fields (AlphaEvolve problem 1). Current research status is unassessed; the repository records AlphaEvolve’s result on this question.

MSC 05B25 · Combinatorial aspects of finite geometries

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository at commit 8f44745 (2026-07-11); the repository records AlphaEvolve results, not current research statusSource & evidence ↓
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Problem statement

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

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Background

Repository result category: world record — AlphaEvolve improved on the best known result recorded at the time. See Section 6.1 in "Mathematical Exploration and Discovery at Scale " paper: arXiv:2511.02864. Experiment notebook.

Progress & source assessment

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

References
  1. [1]Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Repository of problems accompanying “Mathematical exploration and discovery at scale” (arXiv:2511.02864), github.com/google-deepmind/alphaevolve_repository_of_problems (commit 8f44745, 11 July 2026). — Kakeya and Nikodym sets in finite fields (AlphaEvolve problem 1) linkApache-2.0 (repository license); problem pages reproduced with attribution.

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