Koh, DDKohSS LEEPham, TTPhamShen, CYCYShen2023-06-062023-06-0620231071-5797https://scholars.lib.ntu.edu.tw/handle/123456789/631840Let 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 [Formula presented] 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.Rectangles; Distances; Multiplicative subgroups; EXPONENTIAL-SUMS; MULTIPLICATIVE SUBGROUPS; VECTOR-SPACES; PRODUCTS; SUBSETS; FIELDS; ERDOS; SETSjournal article10.1016/j.ffa.2022.1021472-s2.0-85144821947WOS:000917432500001https://scholars.lib.ntu.edu.tw/handle/123456789/631778