Distinct distances in the complex plane

06/16/2020

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by   Adam Sheffer, et al.

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We prove that if P is a set of n points in ℂ^2, then either the points in P determine Ω(n^1-ϵ) complex distances, or P is contained in a line with slope ± i. If the latter occurs then each pair of points in P have complex distance 0.