Distribution of similar configurations in subsets of $\mathbb{F}_q^d$

Abstract: Let $\mathbb{F}q$ be a finite field of order $q$ and $E$ be a set in $\mathbb{F}_q^d$. The distance set of $E$ is defined by $\Delta(E):={\lVert x-y \rVert :x,y\in E}$, where $\lVert \alpha \rVert=\alpha_1^2+\dots+\alpha_d^2$. Iosevich, Koh and Parshall (2018) proved that if $d\geq 2$ is even and $|E|\geq 9q^{d/2}$, then $$\mathbb{F}_q= \frac{\Delta(E)}{\Delta(E)}=\left{\frac{a}{b}: a\in \Delta(E),\ b\in \Delta(E)\setminus{0} \right}.$$ In other words, for each $r\in \mathbb{F}_q^*$ there exist $(x,y)\in E^2$ and $(x',y')\in E^2$ such that $\lVert x-y\rVert\neq0$ and $\lVert x'-y' \rVert=r\lVert x-y\rVert$. Geometrically, this means that if the size of $E$ is large, then for any given $r \in \mathbb{F}_q^*$ we can find a pair of edges in the complete graph $K{|E|}$ with vertex set $E$ such that one of them is dilated by $r\in \mathbb{F}q^*$ with respect to the other. A natural question arises whether it is possible to generalize this result to arbitrary subgraphs of $K{|E|}$ with vertex set $E$ and this is the goal of this paper. In this paper, we solve this problem for $k$-paths $(k\geq 2)$, simplexes and 4-cycles. We are using a mix of tools from different areas such as enumerative combinatorics, group actions and Tur\'an type theorems.