New Nikodym set constructions over finite fields

Terence Tao

Vol. 345 (2026), No. 1, 169–186

DOI: 10.2140/pjm.2026.345.169

Abstract

For any fixed dimension $d\ge 3$ we construct a Nikodym set in ${\mathbb{𝔽}}_q^d$ of cardinality $q^d-\left(\frac{d-2}{\log <!--FUNCTION\; APPLICATION-->2}+1+o(1)\right)q^{d-1}\log <!--FUNCTION\; APPLICATION-->q$ in the limit $q\to \infty$, when $q$ is an odd prime power. This improves upon the naive random construction, which gives a set of cardinality $q^d-(d-1+o(1))q^{d-1}\log <!--FUNCTION\; APPLICATION-->q$, and is new in the regime where ${\mathbb{𝔽}}_q$ has unbounded characteristic and $q$ not a perfect square. While the final proofs are completely human generated, the initial ideas of the construction were inspired by output from the tools AlphaEvolve and DeepThink. We also present a simple construction of Nikodym sets in ${\mathbb{𝔽}}_q^2$ for $q$ a perfect square that is a special case of known unital-based constructions, and matches the existing bounds of $q^2-q^{3∕2}+O(q\log <!--FUNCTION\; APPLICATION-->q)$, assuming that $q$ is not the square of a prime $p\equiv 3(\mathrm{mod}<!--FUNCTION\; APPLICATION-->4)$.

Keywords

Nikodym set, finite fields

Mathematical Subject Classification

Primary: 05B25, 52C35

Milestones

Received: 12 November 2025

Revised: 1 July 2026

Accepted: 1 July 2026

Published: 25 September 2026