Bisector Energy and Few Distinct Distances

Abstract

We define the bisector energy $\mathcal E(\mathcal P)$ of a set $\mathcal P$ in $\mathbb R^2$ to be the number of quadruples $(a,b,c,d)\in \mathcal P^4$ such that a, b determine the same perpendicular bisector as c, d. Equivalently, $\mathcal E(\mathcal P)$ is the number of isosceles trapezoids determined by $\mathcal P$. We prove that for any $\varepsilon >0$, if an n-point set $\mathcal P$ has no M(n) points on a line or circle, then we have

$$\begin{aligned} \mathcal E(\mathcal P) = O\big (M(n)^{\frac{2}{5}}n^{\frac{12}{5}+\varepsilon } + M(n)n^2\big ). \end{aligned}$$

We derive the lower bound $\mathcal E(\mathcal P)=\Omega (M(n)n^2)$, matching our upper bound when M(n) is large. We use our upper bound on $\mathcal E(\mathcal P)$ to obtain two rather different results:

• (i) If $\mathcal P$ determines $O(n/\sqrt{\log n})$ distinct distances, then for any $0<\alpha \le 1/4$, there exists a line or circle that contains at least $n^\alpha$ points of $\mathcal P$, or there exist $\Omega (n^{8/5-12\alpha /5-\varepsilon })$ distinct lines that contain $\Omega (\sqrt{\log n})$ points of $\mathcal P$. 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 $\mathcal P$, the number of distinct perpendicular bisectors determined by $\mathcal P$ is $\Omega \left( \min \left\{ M(n)^{-2/5}n^{8/5-\varepsilon }, M(n)^{-1} n^2\right\} \right)$.