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Finite field Nikodym problem for spread line sets

Ting-Wei Chao, Hung-Hsun Hans Yu

TL;DR

The paper investigates the finite-field Nikodym problem and its connection to Kakeya geometry. It introduces the concept of algebraic spreadness for a set of lines associated with a weak Nikodym set and proves a conditional lower bound: if this line family is algebraically spread, then any weak Nikodym set $N$ in $\mathbb{F}_q^d$ satisfies $|N|\ge q^d-C_d q^{d-1/d}$ for $d\ge3$. The main technical framework uses Hasse derivatives and a polynomial vanishing method together with a dimension-counting argument to bound the size of the line family, yielding a strong incidence-geometric constraint under the spreadness hypothesis. The results provide concrete evidence for the conjectured near-full-size behavior of weak Nikodym sets and connect the Nikodym problem to algebraic properties of line configurations, with potential implications for related finite-field incidence problems.

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.

Finite field Nikodym problem for spread line sets

TL;DR

The paper investigates the finite-field Nikodym problem and its connection to Kakeya geometry. It introduces the concept of algebraic spreadness for a set of lines associated with a weak Nikodym set and proves a conditional lower bound: if this line family is algebraically spread, then any weak Nikodym set in satisfies for . The main technical framework uses Hasse derivatives and a polynomial vanishing method together with a dimension-counting argument to bound the size of the line family, yielding a strong incidence-geometric constraint under the spreadness hypothesis. The results provide concrete evidence for the conjectured near-full-size behavior of weak Nikodym sets and connect the Nikodym problem to algebraic properties of line configurations, with potential implications for related finite-field incidence problems.

Abstract

A set of points is a Nikodym set if, for any , there is a line through such that . We conjecture that and prove it under an extra algebraic assumption.
Paper Structure (4 sections, 8 theorems, 36 equations)

This paper contains 4 sections, 8 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Algebraic spreadness
  4. Main proof

Key Result

Theorem 1.2

For any $d\in\mathbb{N}$, there exists a constant $C_d>0$ so that the following holds. Let $N$ be a weak Nikodym set in $\mathbb{F}^d_q$. Let $\mathcal{L}_N = \{\ell_x:x\in \mathbb{F}^d_q\backslash N\}$ be a set of lines associated with the weak Nikodym set $N$. Let $\mathbb{F}$ be any field extensi

Theorems & Definitions (26)

  • Conjecture 1.1
  • Theorem 1.2
  • Definition 2.1: Hasse derivatives
  • Definition 2.2: Multiplicity
  • Definition 2.3: Restriction of a polynomial
  • Definition 2.4: Multiplicity on a line
  • Proposition 2.5
  • Proposition 2.6
  • Definition 3.1: Algebraic spreadness for points
  • Example 3.2
  • ...and 16 more