Interpolation of point configurations in the discrete plane

Defining distances over finite fields formally by $||x-y||:=(x_1-y_1)^2+\cdots + (x_d-y_d)^2$ for $x,y\in \mathbb{F}_q^d$, distance problems naturally arise in analogy to those studied by Erd\H{o}s and Falconer in Euclidean space. Given a graph $G$ and a set $E\subseteq \mathbb{F}_q^2$, let $\Delta_G(E)$ 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 $|E|\geq q^{d-\frac{d-1}{k+1}}$, it follows that $|\Delta_G(E)|\geq cq^{\binom{k+1}{2}}$. In the case when $k=d=2$, the threshold can be improved to $|E|\geq q^{\frac{8}{5}}$. Moreover, Jardine, Iosevich, and McDonald showed that in the case when $G$ is a tree with $k+1$ vertices, then whenever $E\subseteq \mathbb{F}_q^d$, $d\geq 2$ satisfies $|E|\geq C_kq^{\frac{d+1}{2}}$, it follows that $\Delta_G(E)=\mathbb{F}_q^k$. 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 $|E|\geq q^{\frac{12}{7}}$, it follows that $|\Delta_G(E)|\geq cq^6$.