Asymptotic fullness of finite-field Nikodym sets

Establish that, for every fixed dimension d, the minimum cardinality Nikodym(d,q) of a Nikodym set in F_q^d satisfies Nikodym(d,q)=q^d-o(q^d) as q tends to infinity.

Background

The paper discusses the minimum cardinality of Nikodym sets in the finite vector space F_qd. A Nikodym set contains a punctured line through every point of F_qd. The cited conjecture asserts that such sets must occupy asymptotically almost all of the ambient space as q grows, for every fixed dimension.

The paper notes that this conjecture is known in dimensions d≤2 and that stronger estimates are available in bounded characteristic, but it remains unresolved in the general setting considered by the conjecture. The paper contributes new upper-bound constructions but does not establish the conjectured asymptotic lower bound.

References

It is conjectured Conjecture 1.2 that \begin{equation}\label{nconj} (d,q) = qd - o(qd). \end{equation}

nconj:

(d,q)=qd−o(qd).(d,q) = q^d - o(q^d).

— New Nikodym set constructions over finite fields  (2511.07721 - Tao, 11 Nov 2025) in Introduction, summary of known bounds, bullet point citing Conjecture 1.2 of Lund, Saraf, and Wolf