VC-Dimension and Distance Chains in $\mathbb{F}_q^d$

Consider a set $X$ and a collection $\mathcal{H}$ of functions from $X$ to $\{0,1\}$. We say that $\mathcal{H}$ shatters a finite set $A \subset X$ if the restriction of $\mathcal{H}$ to $A$ yields all $2^{|A|}$ possible functions from $A$ to $\{0,1\}$. The Vapnik-Chervonenkis (VC) dimension of $\mathcal{H}$ is the size of the largest set it shatters. Recent work by Fitzpatrick, Wyman, and the fourth and seventh authors studied the VC-dimension of a natural family of classifiers $\Hh_t^d$ on subsets of $\F_q^d$, the $d$-dimensional vector space over the field $\F_q$. For fixed $t \in \F_q^d$, let $h_y(x) = 1$ if $||y-x||=t$ and 0 otherwise, where $||x|| = x_1^2 + x_2^2+ \cdots + x_d^2$. For $E\subset \F_q^d$, they defined $\Hh_t^d(E) = \{h_y: y \in E\}$ and asked how large $E$ must be to guarantee that the VC-dimension of $\Hh_t^d(E)$ equals that of $\Hh_t^d(\F_q^d)$. They were able to show that this is the case when $|E|\geq Cq^{\frac{15}{8}}$ for $d=2$ for $C$ constant. In this paper we study what happens if we add another parameter to $h_y$ and ask the same question. Specifically, for $y\neq z$ we define $h_{y,z}(x) = 1$ if $||y-x|| = ||z-x|| = t$ and 0 otherwise, and let $\Hh_t^d(E) = \{h_{y,z}: y,z \in E, y\neq z\}$. We resolve this problem in all dimensions, proving that whenever $|E|\geq Cq^{d-\frac{1}{d-1}}$ for $d\geq 3$, the VC-dimension of $\Hh_t^d(E)$ is equal to that of $\Hh_t^d(\F_q^d)$. For the case of $d=3$, we get a slightly stronger result: this result holds as long as $|E|\geq Cq^{7/3}$. Furthermore, when $d=2$ the result holds when $|E|\geq Cq^{7/4}$. Here, $C$ is a function of $d$ and thus constant with respect to $q$.