On the Erdos distinct distance problem in the plane
Abstract
In this paper, we prove that a set of $N$ points in ${\bf R}^2$ has at least $c{N \over \log N}$ distinct distances, thus obtaining the sharp exponent in a problem of Erdös. We follow the setup of Elekes and Sharir which, in the spirit of the Erlangen program, allows us to study the problem in the group of rigid motions of the plane. This converts the problem to one of pointline incidences in space. We introduce two new ideas in our proof. In order to control points where many lines are incident, we create a cell decompostion using the polynomial ham sandwich theorem. This creates a dichotomy: either most of the points are in the interiors of the cells, in which case we immediately get sharp results, or alternatively the points lie on the walls of the cells, in which case they are in the zero set of a polynomial of suprisingly low degree, and we may apply the algebraic method. In order to control points where only two lines are incident, we use the flecnode polynomial of the Rev. George Salmon to conclude that most of the lines lie on a ruled surface. Then we use the geometry of ruled surfaces to complete the proof.
 Publication:

arXiv eprints
 Pub Date:
 November 2010
 arXiv:
 arXiv:1011.4105
 Bibcode:
 2010arXiv1011.4105G
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Classical Analysis and ODEs
 EPrint:
 37 pages. This is a revised version in response to referee comments. The exposition is expanded and some errors corrected