New Nikodym set constructions over finite fields
Terence Tao
TL;DR
The paper advances Nikodym set constructions in $\mathbf{F}_q^d$ for fixed $d\ge3$ by establishing a new upper bound $\operatorname{Nikodym}(d,q) \le q^d - (\frac{d-2}{\log 2}+1+o(1))q^{d-1}\log q$ as $q\to\infty$ with odd $q$, improving on naive random constructions in unbounded-characteristic regimes. The authors couple probabilistic methods with a robust deletion/addition scheme, removing multiple quadratic varieties and adding back a small random set to achieve a robust almost Nikodym set, then upgrade to a full Nikodym set via a second-moment analysis of line-intersection events. A parallel two-dimensional construction recovers the known bound in the square-field case and situates the result within unital-based frameworks. The work highlights the landscape where AI-assisted heuristics guided initial exploration, but the final arguments are human-crafted, contributing new techniques in finite-field incidence geometry with potential implications for related Kakeya/Nikodym problems.
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 $p \equiv 3 \pmod{4}$.