The VC-Dimension and Point Configurations in ${\mathbb F}_q^2$

Abstract

Let X be a set and ${\mathcal H}$ a collection of functions from X to $\{0,1\}$. We say that ${\mathcal H}$ shatters a finite set $C \subset X$ if the restriction of ${\mathcal H}$ yields every possible function from C to $\{0,1\}$. The VC-dimension of ${\mathcal H}$ is the largest number d such that there exists a set of size d shattered by ${\mathcal H}$, and no set of size $d+1$ is shattered by ${\mathcal H}$. Vapnik and Chervonenkis introduced this idea in the early 70s in the context of learning theory, and this idea has also had a significant impact on other areas of mathematics. In this paper we study the VC-dimension of a class of functions ${\mathcal H}$ defined on ${\mathbb F}_q^d$, the d-dimensional vector space over the finite field with q elements. Define

$$ {\mathcal H}^d_t=\{h_y(x):y \in {\mathbb F}_q^d \},$$

where for $x\in {\mathbb F}_q^d$, $h_y(x)=1$ if $\Vert x-y\Vert =t$, and 0 otherwise, where here, and throughout, $\Vert x\Vert =x_1^2+x_2^2+\cdots +x_d^2$. Here $t\in {\mathbb F}_q$, $t\ne 0$. Define ${\mathcal H}_t^d(E)$ the same way with respect to $E \subset {\mathbb F}_q^d$. The learning task here is to find a sphere of radius t centered at some point $y\in E$ unknown to the learner. The learning process consists of taking random samples of elements of E of sufficiently large size. We are going to prove that when $d=2$, and $|E|\geqslant Cq^{{15}/{8}}$, the VC-dimension of ${\mathcal H}^2_t(E)$ is equal to 3. This leads to an intricate configuration problem which is interesting in its own right and requires a new approach.