VC-Dimension of Hyperplanes over Finite Fields

TL;DR

This paper investigates the VC-dimension of hyperplanes over finite fields, generalizing previous results from three dimensions to arbitrary dimensions and improving bounds in three dimensions.

Contribution

It extends the understanding of VC-dimension for hyperplanes over finite fields to all dimensions and refines bounds specifically for three-dimensional cases.

Findings

VC-dimension determined for arbitrary dimensions

Improved bounds for the three-dimensional case

Generalization of previous results to higher dimensions

Abstract

Let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field with $q$ elements. For a subset $E\subseteq \mathbb{F}_q^d$ and a fixed nonzero $t\in \mathbb{F}_q$, let $\mathcal{H}_t(E)=\{h_y: y\in E\}$, where $h_y$ is the indicator function of the set $\{x\in E: x\cdot y=t\}$. Two of the authors, with Maxwell Sun, showed in the case $d=3$ that if $|E|\geq Cq^{\frac{11}{4}}$ and $q$ is sufficiently large, then the VC-dimension of $\mathcal{H}_t(E)$ is 3. In this paper, we generalize the result to arbitrary dimension and improve the exponent in the case $d=3$.