Distinct distances in the complex plane

Sheffer, Adam; Zahl, Joshua

doi:10.1090/tran/8420

Distinct distances in the complex plane
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by Adam Sheffer and Joshua Zahl PDF

Trans. Amer. Math. Soc. 374 (2021), 6691-6725 Request permission

Abstract:

We prove that if $P$ is a set of $n$ points in $\mathbb {C}^2$, then either the points in $P$ determine $\Omega (n^{1-\varepsilon })$ 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.

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