Abstract
We introduce the bisector energy of an npoint set P in the real plane, defined as the number of quadruples (a,b,c,d) from P such that a and b determine the same perpendicular bisector as c and d. If no line or circle contains M(n) points of P, then we prove that the bisector energy is O(M(n)^{2/5}n^{12/5} + M(n)n^2). We also prove the lower bound M(n)n^2, which matches our upper bound when M(n) is large. We use our upper bound on the bisector energy to obtain two rather different results:
(i) If P determines O(n / sqrt(log n)) distinct distances, then for any 0 < a < 1/4, either there exists a line or circle that contains n^a points of P, or there exist n^{8/5  12a/5} distinct lines that contain sqrt(log n) points of P. This result provides new information on a conjecture of ErdÃ¶s regarding the structure of point sets with few distinct distances.
(ii) If no line or circle contains M(n) points of P, then the number of distinct perpendicular bisectors determined by P is min{M(n)^{2/5}n^{8/5}, M(n)^{1}n^2}). This appears to be the first higherdimensional example in a framework for studying the expansion properties of polynomials and rational functions over the real numbers, initiated by Elekes and Ronyai.
BibTeX  Entry
@InProceedings{lund_et_al:LIPIcs:2015:5108,
author = {Ben Lund and Adam Sheffer and Frank de Zeeuw},
title = {{Bisector Energy and Few Distinct Distances}},
booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)},
pages = {537552},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897835},
ISSN = {18688969},
year = {2015},
volume = {34},
editor = {Lars Arge and J{\'a}nos Pach},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5108},
URN = {urn:nbn:de:0030drops51086},
doi = {10.4230/LIPIcs.SOCG.2015.537},
annote = {Keywords: Combinatorial geometry, distinct distances, incidence geometry}
}
Keywords: 

Combinatorial geometry, distinct distances, incidence geometry 
Seminar: 

31st International Symposium on Computational Geometry (SoCG 2015) 
Issue Date: 

2015 
Date of publication: 

11.06.2015 