VC-dimension of Salem sets over finite fields

TL;DR

This work develops a unified Fourier-analytic framework to study VC-dimension for hypothesis classes defined by translates of geometric sets over finite fields, extending distance-geometry questions beyond circles to general algebraic sets and random configurations. A central contribution is a shattering theorem: if a symmetric Salem-type set $S$ satisfies size and intersection bounds and the ambient set $E$ is sufficiently large, then the associated class $\mathcal{H}_S(E)$ has VC-dimension at least 3, with specific conditions giving equality to 3 and no 4-point shattering. The paper also provides explicit Salem-set examples (sphere, paraboloid, and quadratic curves), analyzes random constructions, and contrasts the Euclidean and finite-field settings through a detailed treatment of the symmetrized parabola, showing a sharp dimensional gap between the two worlds. Overall, the results broaden the applicability of VC-dimension methods to structured and random geometric configurations in finite fields, offering tools for probing Erdős-type distance problems and related combinatorial questions. The framework highlights how Fourier decay, intersection properties, and algebraic structure jointly control shattering phenomena with potential extensions to fractal-like sets and higher-degree curves.

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 Erdős distance problem and the Falconer problem when considering a hypothesis class defined by spheres. In particular, the Erdős 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.

Figures (6)

• Figure 1: The graph $G_3$ represents shattering 3 points.
• Figure 2: The graph $G_2$ represents shattering 2 points.
• Figure 3: Left: The cube graph $Q_3$, as constructed by Lemma \ref{['rhombus']} and Corollary \ref{['cube']}. Right: Re-labeling the vertices to agree with Figure \ref{['vc3_graph']}, and deleting the vertex $x^4+v$ which is unnecessary for the proof of Theorem \ref{['shatter3']}.
• Figure 4: Four points placed on the Symmetrized Parabola, used to illustrate the different geometric configurations examined in the casework that follows
• Figure 5: Geometric configuration used in proof of Lemma \ref{['main_case']}, Case 1, where the translates $y^{123}+P$ and $y^{124}+P$ force $x^3,x^4$ to lie to the right of $x^1,x^2$.

Theorems & Definitions (83)

• Theorem 1.1: Guth-Katz
• Definition 1.2
• Remark 1.3
• Theorem 1.4: Iosevich-Rudnev
• Definition 1.5
• Definition 1.6
• Definition 1.7
• Definition 1.8
• Remark 1.9
• Definition 1.10