Interpolation of point configurations in the discrete plane

Defining distances over finite fields formally by $\mathrm{\mid }\mathrm{\mid }x-y\mathrm{\mid }\mathrm{\mid }:=(x_1-y_1{)}^2+\cdots +(x_d-y_d{)}^2$ for $x,y\in {\mathbb{F}}_q^d$, distance problems naturally arise in analogy to those studied by Erdős and Falconer in Euclidean space. Given a graph $G$ and a set $E\subseteq {\mathbb{F}}_q^2$, let ${\mathrm{\Delta }}_G(E)$ 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 $\mathrm{\mid }E\mathrm{\mid }\ge q^{d-\frac{d-1}{k+1}}$, it follows that $\mathrm{\mid }{\mathrm{\Delta }}_G(E)\mathrm{\mid }\ge c{q}^{\left(\genfrac{}{}{0px}{}{k+1}{2}\right)}$. In the case when $k=d=2$, the threshold can be improved to $\mathrm{\mid }E\mathrm{\mid }\ge q^{\frac{8}{5}}$. Moreover, Jardine, Iosevich, and McDonald showed that in the case when $G$ is a tree with $k+1$ vertices, then whenever $E\subseteq {\mathbb{F}}_q^d$, $d\ge 2$ satisfies $\mathrm{\mid }E\mathrm{\mid }\ge C_k{q}^{\frac{d+1}{2}}$, it follows that ${\mathrm{\Delta }}_G(E)={\mathbb{F}}_q^k$. 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 $E\subseteq {\mathbb{F}}_q^2$, $q=p^n$, $n$ odd, $p\equiv 3\text{ mod }4$, and $G$ is the graph consisting of two triangles joined at a vertex, then whenever $\mathrm{\mid }E\mathrm{\mid }\ge q^{\frac{12}{7}}$, it follows that $\mathrm{\mid }{\mathrm{\Delta }}_G(E)\mathrm{\mid }\ge c{q}^6$.