Configurations of rectangles in [formula omitted

Abstract

Let Fq be a finite field of order q. In this paper, we study the distribution of rectangles in a given set in Fq2. More precisely, for any 0<δ≤1, we prove that there exists an integer q0=q0(δ) with the following property: if q≥q0 and A is a multiplicative subgroup of Fq⁎ with |A|≥q2/3, then any set S⊂Fq2 with |S|≥δq2 contains at least ≫|S|4|A|2q5 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.