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arXiv · 2402.15484

On Distinct Angles in the Plane

Sergei V. Konyagin·Jonathan Passant·Misha Rudnev
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Abstract

We prove that if $N$ points lie in convex position in the plane then they determine $\Omega(N^{5/4})$ distinct angles, provided that the points do not lie on a common circle. This is derived from a more general claim that if $N$ points in the convex position in the real plane determine $KN$ distinct angles, then $K=\Omega(N^{1/4})$ or $\Omega(N/K)$ points are co-circular. The proof makes use of the implicit order one can give to points in convex position and relies on a slightly more general order assumption. The assumption enables one to reduce the issue to counting incidences between points and a multiset of cubic curves, with special attention being paid to the case when the curves are reducible.

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Sergei V. Konyagin, Jonathan Passant, Misha Rudnev. 2024-02-23. On Distinct Angles in the Plane. https://arxiv.org/abs/2402.15484

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