论文标题
一般位置的独特角度
Distinct Angles in General Position
论文作者
论文摘要
ERDS独特的距离问题是离散几何形状中无处不在的问题。 ErdőS独特的角度问题的知之甚少，这是在飞机上找到$ N $非校准点之间最小数量的不同角度的问题。最近的工作在此问题的各种变体上引入了界限，灵感来自距离设置中的类似变体。 在此简短说明中，我们改善了最低的上限，以最小数量的不同角度数量，从$ o（n^{\ log_2（7）}）$从$ o（N^{\ log_2（7）}）$形成的不同角度数量到$ O（n^2）$。在这项工作之前，类似的界限依赖于从更高维空间上的通用平面投影。在本文中，我们采用了对数螺旋的几何特性，避开了对投影的需求。 我们还使用此配置来减少最大整数上的上限，以使任何一组$ n $点的$ n $点具有所有不同角度的子集。此键从$ O（n^{\ log_2（7）/3}）$降低到$ O（n^{1/2}）$。
The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is Erdős' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the plane. Recent work has introduced bounds on a wide array of variants of this problem, inspired by similar variants in the distance setting. In this short note, we improve the best known upper bound for the minimum number of distinct angles formed by $n$ points in general position from $O(n^{\log_2(7)})$ to $O(n^2)$. Before this work, similar bounds relied on projections onto a generic plane from higher dimensional space. In this paper, we employ the geometric properties of a logarithmic spiral, sidestepping the need for a projection. We also apply this configuration to reduce the upper bound on the largest integer such that any set of $n$ points in general position has a subset of that size with all distinct angles. This bound is decreased from $O(n^{\log_2(7)/3})$ to $O(n^{1/2})$.
