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Let $\mathbb{F}_q$ be a finite field of order $q$. In this paper, we studythe distribution of rectangles in a given set in $\mathbb{F}_q^2$. Moreprecisely, for any $0<\delta\le 1$, we prove that there exists an integer$q_0=q_0(\delta)$ with the following property: if $q\ge q_0$ and $A$ is amultiplicative subgroup of $\mathbb{F}^*_q$ with $|A|\ge q^{2/3}$, then any set$S\subset \mathbb{F}_q^2$ with $|S|\ge \delta q^2$ contains at least $\gg\frac{|S|^4|A|^2}{q^5}$ rectangles with side-lengths in $A$. We also considerthe case of rectangles with one fixed side-length and the other in amultiplicative subgroup $A$.

Configurations of rectangles in a set in $\mathbb{F}_q^2$