VC-dimension of Salem sets over finite fields

TL;DR

This paper investigates the VC-dimension of geometric hypothesis classes over finite fields, extending existing distance problem techniques to a broader context including algebraic and random sets.

Contribution

It introduces a unified framework for analyzing VC-dimension over finite fields that applies to various structured and random sets, beyond traditional geometric objects.

Findings

Techniques for distance problems extend to broader contexts.

Unified framework for VC-dimension analysis.

Applicable to algebraic curves and random sets.

Abstract

The VC-dimension, introduced by Vapnik and Chervonenkis in 1968 in the context of learning theory, has in recent years provided a rich source of problems in combinatorial geometry. Given $E\subseteq \mathbb{F}_q^d$ or $E\subseteq \mathbb{R}^d$, finding lower bounds on the VC-dimension of hypothesis classes defined by geometric objects such as spheres and hyperplanes is equivalent to constructing appropriate geometric configurations in $E$. The complexity of these configurations increases exponentially with the VC-dimension. These questions are related to the Erd\H{o}s distance problem and the Falconer problem when considering a hypothesis class defined by spheres. In particular, the Erd\H{o}s distance problem over finite fields is equivalent to showing that the VC-dimension of translates of a sphere of radius $t$ is at least one for all nonzero $t\in \mathbb{F}_q$. In this paper, we show that many of the existing techniques for distance problems over finite fields can be extended to a much broader context, not relying on the specific geometry of circles and spheres. We provide a unified framework which allows us to simultaneously study highly structured sets such as algebraic curves, as well as random sets.