Given a set of points P ? F q 2 such that | P | ? q 4 / 3 , we establish that for a positive proportion of points a ? P , we have | { ? a - b ? : b ? P } | ? q , where ? 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 | ? q 3 / 2 , then the number B of distinct lines which arise as the perpendicular bisector of two points in P satisfies B ? q 2 .
['Brandon Hanson', 'Ben Lund', 'Oliver Roche-Newton']
On distinct perpendicular bisectors and pinned distances in finite fields