The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. 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. The standard problem is already well understood. However, it admits many of the same variants as the distinct distance problem, many of which are unstudied. We provide upper and lower bounds on a broad class of distinct angle problems. We show that the number of distinct angles formed by $n$ points in general position is $O(n^{\log_2(7)})$, providing the first non-trivial bound for this quantity. We introduce a new class of asymptotically optimal point configurations with no four cocircular points. Then, we analyze the sensitivity of asymptotically optimal point sets to perturbation, yielding a much broader class of asymptotically optimal configurations. In higher dimensions we show that a variant of Lenz's construction admits fewer distinct angles than the optimal configurations in two dimensions. We also show that the minimum size of a maximal subset of $n$ points in general position admitting only unique angles is $\Omega(n^{1/5})$ and $O(n^{\log_2(7)/3})$. We also provide bounds on the partite variants of the standard distinct angle problem.
翻译:Erd\ H{o} 不同的距离问题是离散几何中一个普遍存在的问题。 不太广为人知的是 Erd\ H{o} 的独特角度问题, 找到在平面上一美元非双线点之间不同角度的最低数量的问题。 标准问题已经非常清楚。 但是, 它承认许多相同的变量与不同的距离问题相同, 其中很多是未经研究的。 我们提供不同角度问题的广泛类别上下限。 我们显示, 一般位置上一美元点形成的不同角度的数量是 $( n ⁇ log_ 2(7)}) 美元, 提供了第一个非三边点约束的这一数量。 我们引入了一个新的非双边最佳点配置类别, 没有四个眼点。 然后, 我们分析了不同时的最佳点的敏感度, 造成更宽的更宽度, 产生更宽的一等的最佳配置。 在更高层面中, lez 美元 1 标准值 1 和 美元 标准值 1 值 的最小值 的 O 位数, 也显示, 我们的最小值 1 和 美元 最低值 水平 。