Bisector Energy and Few Distinct Distances

We define the bisector energyE(P) of a set P in R^2 to be the number of quadruples (a,b,c,d)∈P^4 such that a, b determine the same perpendicular bisector as c, d. Equivalently, E(P) is the number of isosceles trapezoids determined by P. We prove that for any ε>0, if an n-point set P has no M(n) points on a line or circle, then we have E(P)=O(M(n)^(2/5)n^(12/5+ε) +M(n)n^2). We derive the lower bound E(P)=Ω(M(n)n^2), matching our upper bound when M(n) is large. We use our upper bound on E(P) to obtain two rather different results: (i) If P determines O(n/√log n) distinct distances, then for any 0<α≤1/4, there exists a line or circle that contains at least n^α points of P, or there exist Ω(n^(8/5−12α/5−ε)) distinct lines that contain Ω(/√log n) points of P. This result provides new information towards a conjecture of Erdős (Discrete Math 60:147–153, 1986) regarding the structure of point sets with few distinct distances. (ii) If no line or circle contains M(n) points of P, the number of distinct perpendicular bisectors determined by P is Ω(min{M(n)^(−2/5)n^(8/5−ε), M(n)^(-1) n^2}).