On Distinct Angles in the Plane
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866912642917990400 |
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| author | Konyagin, Sergei V. Passant, Jonathan Rudnev, Misha |
| author_facet | Konyagin, Sergei V. Passant, Jonathan Rudnev, Misha |
| contents | We prove that if $N$ points lie in convex position in the plane then they determine $Ω(N^{5/4})$ 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 $K=Ω(N^{1/4})$ or $Ω(N/K)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15484 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Distinct Angles in the Plane Konyagin, Sergei V. Passant, Jonathan Rudnev, Misha Combinatorics Metric Geometry 52C10 We prove that if $N$ points lie in convex position in the plane then they determine $Ω(N^{5/4})$ 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 $K=Ω(N^{1/4})$ or $Ω(N/K)$ 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. |
| title | On Distinct Angles in the Plane |
| topic | Combinatorics Metric Geometry 52C10 |
| url | https://arxiv.org/abs/2402.15484 |