论文标题
一个三角形的结构定理
A Structural Theorem for Sets With Few Triangles
论文作者
论文摘要
我们表明，如果有限点集$ p \ subseteq \ mathbb {r}^2 $具有最少的三角形类别类别，则最大为$ m $，然后至少一个以下一个。 （1）有一个$σ> 0 $和一个$ω（| p |^σ）$点$ p $的$ l $。此外，$ p $的正比例由平行于$ω（| p |^σ）$点的$ p $ $ l $覆盖。 （2）有一个圆$γ$，其中包含$ p $的正比例。 这为ERDS的两个猜想提供了证据。我们将Petridis-Roche-Newton-Rudnev-Warren的结果与仿射组的结构结合在一起，并结合了添加剂组合学的经典结果。
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 $σ>0$ and a line $l$ which contains $Ω(|P|^σ)$ points of $P$. Further, a positive proportion of $P$ is covered by lines parallel to $l$ each containing $Ω(|P|^σ)$ points of $P$. (2) There is a circle $γ$ which contains a positive proportion of $P$. This provides evidence for two conjectures of Erdős. We use the result of Petridis-Roche-Newton-Rudnev-Warren on the structure of the affine group combined with classical results from additive combinatorics.
