典型避免单位距离集合中的聚类现象
Clustering in typical unit-distance avoiding sets
AI总结：
本文在平面中严格证明，稠密乃至典型的避免单位距离集合都会出现距离约为2的点对过度富集的聚类现象，并基于线性规划方法加以论证。
摘要：
20世纪60年代，Moser提出问题：如果$\n\mathbb{R}^d$的一个子集中不存在任何两点恰好相距1，那么该子集的密度可以有多大。长期以来已有一系列研究给出了这种密度的上界。稠密的避免单位距离集合有一个令人好奇的特征：它们似乎是“成块的”，也就是说，禁止单位距离与拥有多于预期数量的距离约为2的点对相伴出现。在本文中，我们在$\n\mathbb{R}^2$中严格确立了这一现象。我们证明，稠密的避免单位距离集合中距离约为2的点对占比过高，并且这种聚类现象扩展到典型的避免单位距离集合。为此，我们在以往用于证明避免单位距离集合密度上界的线性规划方法基础上进行构建。
英文摘要：
In the 1960s Moser asked how dense a subset of $\mathbb{R}^d$ can be if no pairs of points in the subset are exactly distance 1 apart. There has been a long line of work showing upper bounds on this density. One curious feature of dense unit distance avoiding sets is that they appear to be ``clumpy,'' i.e. forbidding unit distances comes hand in hand with having more than the expected number distance $\approx 2$ pairs. In this work we rigorously establish this phenomenon in $\mathbb{R}^2$. We show that dense unit distance avoiding sets have over-represented distance $\approx 2$ pairs, and that this clustering extends to typical unit distance avoiding sets. To do so, we build off of the linear programming approach used previously to prove upper bounds on the density of unit distance avoiding sets.
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