The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is Erd\H{o}s' 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.

翻译：Erd\H{o} 不同的距离问题是离散几何中普遍存在的问题。 不太广为人知的是, Erd\H{o} 的独特角度问题, 即如何在平面上找到一美元非双向点之间最小不同角度的问题。 最近的工作在距离设置中类似的变量的启发下, 引入了这一问题各种变量的界限 。 在这个简短的注释中, 我们改进了以一般位置一美元点形成的最低不同角度数量为最知名的上限, 从$O( n ⁇ log_ 2(7)}) 美元到$O( n ⁇ 2) 美元。 在这项工作之前, 类似的界限依赖于从高维空间投射到通用平面的投影。 在本文中, 我们使用对数螺旋的几何特性, 以投影的必要性 。