Interpolation of point configurations in the discrete plane

Defining distances over finite fields formally by ||x-y||:=(x₁-y₁)²+⋯ + (xd-yd)² for x,y∈ Fqᵈ, distance problems naturally arise in analogy to those studied by Erd{o}s and Falconer in Euclidean space. Given a graph G and a set E⊆ Fq², let ΔG(E) be the generalized distance set corresponding to G. In the case when G is the complete graph on $k+1$ vertices, Bennett, Hart, Iosevich, Pakianathan, and Rudnev showed that when |E|≥ qd-d-1/k+1, it follows that |ΔG(E)|≥ cq^k+12. In the case when $k=d=2$, the threshold can be improved to |E|≥ q8/5. Moreover, Jardine, Iosevich, and McDonald showed that in the case when G is a tree with $k+1$ vertices, then whenever E⊆ Fqᵈ, d≥ 2 satisfies |E|≥ Cₖqd+1/2, it follows that ΔG(E)=Fqᵏ. In this paper, we present a technique which enables us to study certain graphs with both rigid and non-rigid components. In particular, we show that for E⊆ Fq², q=pⁿ, n odd, p≡ 3 \ mod \ 4, and G is the graph consisting of two triangles joined at a vertex, then whenever |E|≥ q12/7, it follows that |ΔG(E)|≥ cq⁶.