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 ${\mathbb{F}}_q^{\ast }$ 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 $A$.

11 pages