Finite field Nikodym problem for spread line sets

Abstract

A set of points $N\subseteq {\mathbb{F}}_q^d$ is a Nikodym set if, for any $x\in {\mathbb{F}}_q^d$, there is a line $\ell$ through $x$ such that $\ell \setminus \{x\}\subseteq N$. We conjecture that $|N|=q^d-O_d(q^{d/(d-1)})$ and prove it under an extra algebraic assumption.

8 pages