# Few distinct distances and perpendicular bisectors

# Few distinct distances and perpendicular bisectors

Let d(n) be the smallest number of distinct distances determined by any set of n points in the real plane. For n sufficiently large, is each set of n points that determines d(n) distances the intersection of an equalateral triangular lattice with a convex set? Is there at least a line that contains n^ε points, for some ε>0? Erdős asked these questions 30 years ago, and an argument due to Szemerédi shows that, if P is a set of points that determines n/k distances, then the perpendicular bisector of some pair of the points must be incident to k of the points. I will present an upper bound on the number of quadruples (x,y,z,w) among a set of points such that the perpendicular bisector of (x,y) is the same as the bisector of (z,w). Combined with Szemerédi's argument, this bound implies that set of n points that determines n/k distinct distances either includes Omega(n^{7/5}) collinear k-tuples, or a single circle or line contains Omega(n^{1/12}) points. Joint work with Adam Sheffer and Frank de Zeeuw.