New Nikodym set constructions over finite fields

TL;DR

This paper introduces improved constructions of Nikodym sets over finite fields for higher dimensions, achieving smaller set sizes than previous methods, especially when the field size has unbounded characteristic.

Contribution

The paper presents new explicit constructions of Nikodym sets in higher dimensions over finite fields that outperform naive random approaches, particularly for fields with unbounded characteristic.

Findings

Constructed Nikodym sets with smaller size bounds in high dimensions.

Provided explicit constructions matching known bounds in two dimensions.

Achieved improvements over random constructions in the unbounded characteristic regime.

Abstract

For any fixed dimension $d \geq 3$ we construct a Nikodym set in $F_q^d$ of cardinality $q^d - (\frac{d-2}{\log 2} +1+o(1)) q^{d-1} \log q$ 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 $q^d - (d-1+o(1)) q^{d-1} \log q$, and is new in the regime where $F_q$ 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 \texttt{AlphaEvolve} and \texttt{DeepThink}. We also present a simple construction of Nikodym sets in $F_q^2$ for $q$ a perfect square that is a special case of known unital-based constructions, and matches the existing bounds of $q^2 - q^{3/2} + O(q \log q)$, assuming that $q$ is not the square of a prime $p \equiv 3 \pmod{4}$.