Distinct distances in the complex plane
Adam Sheffer, Joshua Zahl · Transactions of the American Mathematical Society · 2021
We prove that if P P is a set of n n points in C 2 \mathbb {C}^2 , then either the points in P P determine Ω ( n 1 − ε ) \Omega (n^{1-\varepsilon }) complex distances, or P P is contained in a line with slope ± i \pm i . If the latter occurs then each pair of points in P P have complex distance 0.