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

Abstract: Let $\mathbb{F}_q$ be a finite field of order $q$. In this paper, we study the distribution of rectangles in a given set in $\mathbb{F}_q^2$. More precisely, 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 a multiplicative 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 consider the case of rectangles with one fixed side-length and the other in a multiplicative subgroup $A$.