Distinct distances in the complex plane

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.

41 pages, 0 figures

References in corpus (1)

• Sphere tangencies, line incidences, and Lie's line-sphere correspondence

Cited by in corpus (3)

• Erdős distinct distances in hyperbolic surfaces
• Distinct distances on hyperbolic surfaces
• Distinct Distances with ${\ell }_p$ Spaces