论文标题
三个维度的不同角度和角链
Distinct Angles and Angle Chains in Three Dimensions
论文作者
论文摘要
1946年，Erdős提出了独特的距离问题，该问题旨在找到从飞机中任何$ n $点的任何配置中选择的点对之间的最小距离数。此后，该问题与许多变体一起探讨了，包括将其扩展到更高维度的变体。研究较少但同样吸引人的是Erdő的独特角度问题，该问题旨在在平面中找到最小化不同角度数量的点配置。在最近的论文“一般位置的不同角度”中，Fleischmann，Konyagin，Miller，Palsson，Pesikoff和Wolf使用对数的螺旋形，以$ O（N^2）$的上限建立了飞机上最小数量的不同角度的上限，这在任何一线或四个线上都在任何圈子上或四个圈子上都预示着。 我们考虑了三个维度的不同角度的问题，并在此环境中以一般位置的最小数量来提供界限。我们专注于固定的问题的变体，并检查了$ \ mathbb {r}^3 $中点配置的显式结构，这些构造利用自相似，以最大程度地减少不同角度的数量。此外，我们研究了有关不同角度链的独特角度问题的变体，并在$ \ mathbb {r}^2 $和$ \ mathbb {r}^3 $中提供了最小链数的界限。
In 1946, Erdős posed the distinct distance problem, which seeks to find the minimum number of distinct distances between pairs of points selected from any configuration of $n$ points in the plane. The problem has since been explored along with many variants, including ones that extend it into higher dimensions. Less studied but no less intriguing is Erdős' distinct angle problem, which seeks to find point configurations in the plane that minimize the number of distinct angles. In their recent paper "Distinct Angles in General Position," Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf use a logarithmic spiral to establish an upper bound of $O(n^2)$ on the minimum number of distinct angles in the plane in general position, which prohibits three points on any line or four on any circle. We consider the question of distinct angles in three dimensions and provide bounds on the minimum number of distinct angles in general position in this setting. We focus on pinned variants of the question, and we examine explicit constructions of point configurations in $\mathbb{R}^3$ which use self-similarity to minimize the number of distinct angles. Furthermore, we study a variant of the distinct angles question regarding distinct angle chains and provide bounds on the minimum number of distinct chains in $\mathbb{R}^2$ and $\mathbb{R}^3$.
