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个点构成的点集中，不同点对之间距离的最小数量。此后，该问题及其众多变体（包括向高维空间的扩展）得到了广泛研究。同样引人入胜但研究较少的是埃尔德什的不同角度问题，它寻求在平面上最小化不同角度数量的点配置。在其近期论文《一般位置下的不同角度》中，弗莱施曼、科尼亚金、米勒、帕尔松、佩西科夫和沃尔夫利用对数螺线，证明了一般位置（即无三点共线或四点共圆）下平面上不同角度数量的最小值的上界为O(n²)。我们探讨了三维空间中的不同角度问题，并给出了该设定下一般位置中不同角度数量最小值的界。我们聚焦于该问题的固定变体，并通过考察Self-similarity策略构造的R³中点配置来最小化不同角度数量。此外，我们还研究了关于不同角度链的变体问题，并给出了R²和R³中不同角度链数量最小值的界。