Asymptotic size of finite field Nikodym sets

Prove that for every fixed dimension d ≥ 1 and prime power q → ∞, the minimal cardinality C^N(d,q) of a Nikodym set in the finite vector space F_q^d satisfies C^N(d,q) = q^d − o(q^d).

Background

Nikodym sets in F_q^d are those in which every point x lies on a line contained in N ∪ {x}. Determining their minimal size is a central problem connected to finite geometry, combinatorics, and analogues of the Kakeya conjecture. Known results include comparability to Kakeya sets and sharp densities for certain cases, but the general asymptotic behavior remains conjectural.

The conjecture cited is widely referenced in the literature due to implications for polynomial method techniques and structural properties of line incidences in finite fields.