Finite field Nikodym problem for spread line sets

TL;DR

This paper investigates the finite field Nikodym problem for spread line sets, proposing a conjecture on the minimal size of Nikodym sets and proving it under an additional algebraic condition.

Contribution

It introduces a conjecture on the size of finite field Nikodym sets and provides a proof under a specific algebraic assumption, advancing understanding in finite geometry.

Findings

Proposes a conjecture on the size of Nikodym sets in finite fields.

Proves the conjecture under an extra algebraic assumption.

Contributes to finite field geometry and combinatorics.

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.