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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 $\Delta(E):=\{\lVertx-y \rVert :x,y\in E\}$, where $\lVert \alpha\rVert=\alpha_1^2+\dots+\alpha_d^2$. Iosevich, Koh and Parshall (2018) provedthat 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 anygiven $r \in \mathbb{F}_q^*$ 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 itis possible to generalize this result to arbitrary subgraphs of $K_{|E|}$ withvertex set $E$ and this is the goal of this paper.  In this paper, we solve this problem for $k$-paths $(k\geq 2)$, simplexes and4-cycles. We are using a mix of tools from different areas such as enumerativecombinatorics, group actions and Tur\'an type theorems.

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