相异与重复点积树的界
Bounds on distinct and repeated dot product trees
AI总结：
受Erdős距离问题启发，研究平面有限点集中加权树对应的点积构型，给出相异点积集数量的新下界，并证明特殊点积集存在大量重复以缩小上下界差距。
摘要：
我们受Erdős著名的距离问题启发，以点积代替距离，且针对不止一对点的情形展开研究。具体而言，我们研究平面中大型有限点集内由加权树描述的点构型。我们给出了任意大型有限点集中，作为给定类型树权重的相异点积集合数量的新下界。我们还通过不同构造证明，给定类型树中出现的某些特殊点积集合存在大量重复，缩小了这些构型已知最优上界与下界之间的差距。
英文摘要：
We study questions inspired by Erd\H os' celebrated distance problems with dot products in lieu of distances, and for more than a single pair of points. In particular, we study point configurations present in large finite point sets in the plane that are described by weighted trees. We give new lower bounds on the number of distinct sets of dot products serving as weights for a given type of tree in any large finite point set. We also as demonstrate the existence of many repetitions of some special sets of dot products occurring in a given type of tree in different constructions, narrowing gap between the best known upper and lower bounds on these configurations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
