Bisector Energy and Few Distinct Distances

We introduce the bisector energy of an n-point set P in R2, defined as E(P) = ∣∣{(a, b, c, d) ∈ P | a, b have the same perpendicular bisector as c, d}∣∣ . If no line or circle contains M(n) points of P, then we prove that for any ε > 0 E(P) = O ( M(n) 2 5n 12 5 +ε +M(n)n ) . We also derive the lower bound E(P) = Ω(M(n)n2), which matches 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, either there exists a line or circle that contains nα points of P, or there exist Ω(n8/5−12α/5−ε) distinct lines that contain Ω( √ log n) points of P. This result provides new information on a conjecture of Erdős [7] 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)−2/5n8/5−ε,M(n)−1n2 }) . This appears to be the first higher-dimensional example in a framework for studying the expansion properties of polynomials and rational functions over R, initiated by Elekes and Rónyai [2]. ∗Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM) in Los Angeles, which is supported by the National Science Foundation. Work on this paper by Frank de Zeeuw was partially supported by Swiss National Science Foundation Grants 200020-144531 and 200021-137574. Work on this paper by Ben Lund was supported by NSF grant CCF-1350572. †Department of Computer Science, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8004. [email protected] ‡Department of Mathematics, California Institute of Technology, 1200 East California Blvd Pasadena, CA 91125. [email protected] §EPFL, Lausanne, Switzerland. [email protected] 1 ar X iv :1 41 1. 68 68 v1 [ m at h. C O ] 2 5 N ov 2 01 4