A Refined Energy Bound for Distinct Perpendicular Bisectors

Abstract

Let P be a set of n points in the Euclidean plane. We prove that, for any ε > 0, either a single line or circle contains n/2 points of P, or the number of distinct perpendicular bisectors determined by pairs of points in P is Ω(n 52/35−ε ), where the constant implied by the Ω notation depends on ε. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of P, or the number of distinct perpendicular bisectors is Ω(n 2 ).The proof relies bounding the size of a carefully selected subset of the quadruples (a, b, c, d) ∈ P 4 such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d.

Bisectors and Pinned Distances

“…If no circle or line contains more than M points of A, then Q M = Q, and we recover Theorem 2. Theorem 3 follows from the arguments in [11] by using Theorem 17 in place of [11,Lemma 11].…”

“…The conjecture of Lund, Sheffer, and de Zeeuw on this question is that, for a set A of N points in R 2 , either N/2 points are contained in a single line or circle, or A determines Ω(N 2 ) distinct perpendicular bisectors. Lund, Sheffer, and de Zeeuw showed that a set of N points, no more than M of which are incident to any line or circle, determines Ω ε (min(M −2/5 N 8/5−ε , M −1 N 2 )) distinct perpendicular bisectors, for any ε > 0, with the implied constant depending on ε. Lund [11] showed that a set of N points in the plane, no more than N/2 of which are incident to any line or circle, determines Ω ε (N 52/35−ε ) distinct perpendicular bisectors, for any ε > 0, with the implied constant depending on ε. A slight generalization of Theorem 2, combined with the arguments of [11], gives the following improvement to Lund's bound.…”

“…Lund, Sheffer, and de Zeeuw showed that a set of N points, no more than M of which are incident to any line or circle, determines Ω ε (min(M −2/5 N 8/5−ε , M −1 N 2 )) distinct perpendicular bisectors, for any ε > 0, with the implied constant depending on ε. Lund [11] showed that a set of N points in the plane, no more than N/2 of which are incident to any line or circle, determines Ω ε (N 52/35−ε ) distinct perpendicular bisectors, for any ε > 0, with the implied constant depending on ε. A slight generalization of Theorem 2, combined with the arguments of [11], gives the following improvement to Lund's bound. This is strictly better than Lund's bound, and improves the bound of Lund, Sheffer, and de Zeeuw for point sets with at most N 1/4 points on any line or circle.…”