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arXiv · 2206.09740

A Structural Theorem for Sets With Few Triangles

Sam Mansfield·Jonathan Passant
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Abstract

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.

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Sam Mansfield, Jonathan Passant. 2022-06-20. A Structural Theorem for Sets With Few Triangles. https://doi.org/10.1007/s00493-023-00066-z

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