The VC-dimension and point configurations in ${Bbb F}_q^2$

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 ${Bbb F}_q^d$, the $d$-dimensional vector space over the finite field with $q$ elements. Define $$ {mathcal H}^d_t={h_y(x): y in {Bbb F}_q^d },$$ where for $x in {Bbb F}_q^d$, $h_y(x)=1$ if $||x-y||=t$, and $0$ otherwise, where here, and throughout, $||x||=x_1^2+x_2^2+dots+x_d^2$. Here $t in {Bbb F}_q$, $t ot=0$. Define ${mathcal H}_t^d(E)$ the same way with respect to $E subset {Bbb F}_q^d$. 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 $|E| ge Cq^{frac{15}{8}}$, 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.