Improved quadratic-variety construction for Nikodym sets

Establish, for every fixed d≥3 and odd prime power q tending to infinity, a Nikodym set in F_q^d with cardinality at most q^d-((d-1)/log 2+o(1))q^{d-1}log q, thereby proving the stronger bound suggested by the quadratic-variety heuristic.

Background

The paper proves a weaker upper bound for odd prime powers: qd-((d-2)/log 2+1+o(1))q{d-1}log q. Earlier heuristic arguments based on deleting random quadratic varieties suggested the stronger coefficient (d-1)/log 2.

The authors state that repeated attempts, including computer-assisted exploration, did not make this stronger heuristic rigorous. They therefore explicitly leave the stronger estimate as an open conjecture. Proving it would improve the construction of small Nikodym sets in dimensions d≥3 in the regime of odd, non-square finite fields as well as other odd prime-power cases.

References

On the other hand, we were unable to obtain the improved bound nik-conj despite the (computer-assisted) heuristic argument suggesting it, and pose it instead as an open conjecture.

nik-conj:

(d,q)≤qd−(d−1log⁡2+o(1))qd−1log⁡q;(d,q) \leq q^d - \left(\frac{d-1}{\log 2}+o(1)\right) q^{d-1} \log q;

— New Nikodym set constructions over finite fields  (2511.07721 - Tao, 11 Nov 2025) in Introduction, immediately after Theorem 1 (New upper bound on Nikodym sets) and before the two-dimensional construction