On distinct perpendicular bisectors and pinned distances in finite fields

Given a set of points P⊂Fq2 such that |P|≥q4/3|P|≥q4/3, we establish that for a positive proportion of points a∈Pa∈P, we have|{‖a−b‖:b∈P}|≫q,|{‖a−b‖:b∈P}|≫q, where ‖a−b‖‖a−b‖ is the distance between points a and b. This improves a result of Chapman et al. [6].A key ingredient of our proof also shows that, if |P|≥q3/2|P|≥q3/2, then the number B of distinct lines which arise as the perpendicular bisector of two points in P   satisfies B≫q2B≫q2.