In 1946, Erd\H{o}s 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 Erd\H{o}s' 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 $\mathbb{R}^3$.

翻译：1946年，埃尔德什提出了不同距离问题，旨在寻找平面上任意$n$点构型中点对之间不同距离的最小数目。该问题及其众多变体（包括扩展到高维空间的版本）此后得到了广泛探索。虽研究较少但同样引人入胜的是埃尔德什的不同角度问题，该问题致力于寻找平面上最小化不同角度数目的点构型。在近期论文《一般位置下的不同角度》中，Fleischmann、Konyagin、Miller、Palsson、Pesikoff和Wolf利用对数螺线建立了平面上一般位置（禁止三点共线或四点共圆）下不同角度最小数目$O(n^2)$的上界。我们考虑三维空间中的不同角度问题，并在此背景下给出一般位置下不同角度最小数目的界。我们聚焦于问题的固定变体，并研究$\mathbb{R}^3$中利用自相似性最小化不同角度数目的点构型显式构造。此外，我们探讨了关于不同角度链的变体问题，并给出了$\mathbb{R}^2$和$\mathbb{R}^3$中不同链最小数目的界。