On the Erdos distinct distance problem in the plane

TL;DR

This paper proves that any set of N points in the plane determines at least cN/ log N distinct distances, achieving the optimal exponent in Erdős's problem by combining algebraic and combinatorial geometry techniques.

Contribution

It introduces a novel combination of polynomial cell decomposition and ruled surface analysis to establish the lower bound on distinct distances in the plane.

Findings

Established the sharp lower bound cN/ log N for distinct distances in the plane.

Developed a new method using polynomial ham sandwich theorem for point-line incidence control.

Applied algebraic geometry, including ruled surface theory, to solve a classical combinatorial geometry problem.

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 Erd\"os. We follow the set-up 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 point-line 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.