Dot products in [formula omitted] and the Vapnik-Chervonenkis dimension

Abstract

Given a set E⊂Fq3, where Fq is the field with q elements. Consider a set of “classifiers” Ht3(E)={hy:y∈E}, where hy(x)=1 if x⋅y=t, x∈E, and 0 otherwise. We are going to prove that if |E|≥Cq114, with a sufficiently large constant C>0, then the Vapnik-Chervonenkis dimension of Ht3(E) is equal to 3. In particular, this means that for sufficiently large subsets of Fq3, the Vapnik-Chervonenkis dimension of Ht3(E) is the same as the Vapnik-Chervonenkis dimension of Ht3(Fq3). In some sense the proof leads us to consider the most complicated possible configuration that can always be embedded in subsets of Fq3 of size ≥Cq114.