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Level B · Reproducible Combinatorics P-kakeya-and-nikodym-sets-in-finite-fields

Kakeya and Nikodym sets in finite fields

Let d ≥ 1, and let q be a prime power. Let 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 𝔽_q^d is contained in a line that is contained in N ∪ x.

From the catalogue. Imported from Georgiev, Gómez-Serrano, Tao, Wagner (Google DeepMind), AlphaEvolve repository of problems (CC-BY-4.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-kakeya-and-nikodym-sets-in-finite-fields,
  title        = {Kakeya and Nikodym sets in finite fields},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/kakeya-and-nikodym-sets-in-finite-fields}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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The problem

The problem

Let , and let be a prime power. Let be a finite field of order . A Kakeya set is a set that contains a line in every direction, and an Nikodym set is a set with the property that every point in is contained in a line that is contained in . Let denote the least size of a Kakeya or Nikodym set in respectively.

Known results

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

AlphaEvolve found a construction better than the previous record (at publication).

What counts as progress

  • A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
  • A proof that a known construction is optimal, or a better bound on the other side.
  • Literature: earlier or newer records (literature claims).

Source and licence

Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.