A Structural Theorem for Sets With Few Triangles
| Autor: | Mansfield, Sam, Passant, Jonathan |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Combinatorica (2023) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00493-023-00066-z |
| Popis: | We show that if a finite point set $P\subseteq \mathbb{R}^2$ has the fewest congruence classes of triangles possible, up to a constant $M$, then at least one of the following holds. (1) There is a $\sigma>0$ and a line $l$ which contains $\Omega(|P|^\sigma)$ points of $P$. Further, a positive proportion of $P$ is covered by lines parallel to $l$ each containing $\Omega(|P|^\sigma)$ points of $P$. (2) There is a circle $\gamma$ which contains a positive proportion of $P$. This provides evidence for two conjectures of Erd\H{o}s. We use the result of Petridis-Roche-Newton-Rudnev-Warren on the structure of the affine group combined with classical results from additive combinatorics. Comment: 18 pages, refereed version |
| Databáze: | arXiv |
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