Distinct distances in the complex plane

TL;DR

This paper establishes a lower bound on the number of distinct complex distances determined by a set of points in the complex plane, with a specific structural exception involving points on a line with slope ±i.

Contribution

It proves a new lower bound on the number of distinct complex distances in the complex plane, characterizing the structure of sets with few distances.

Findings

Sets in $ ext{C}^2$ determine $ ext{Omega}(n^{1- ext{epsilon}})$ distances unless all points lie on a line with slope ±i.

If points lie on such a line, all pairwise distances are zero.

The result extends classical distance problems to the complex setting.

Abstract

We prove that if $P$ is a set of $n$ points in $\mathbb{C}^2$, then either the points in $P$ determine $\Omega(n^{1-\epsilon})$ complex distances, or $P$ is contained in a line with slope $\pm i$. If the latter occurs then each pair of points in $P$ have complex distance 0.