典型范数中的单位距离与不同距离
Unit and distinct distances in typical norms
AI总结：
该论文在Baire范畴意义上对$\nmathbb{R}^d$上几乎所有范数解决了Erdős单位距离和不同距离问题，给出紧确界限，并证明几乎所有平面范数的单位距离图色数为4。
摘要：
Erdős的单位距离问题和Erdős的不同距离问题是离散数学中最经典且广为人知的开放问题之一。它们分别询问在欧几里得平面中由$n$个点确定的单位距离的最大数量，或不同距离的最小数量。如果考虑除欧几里得平面以外的赋范空间，这些问题会发生什么变化，这一问题由Ulam和Erdős在1980年代提出，并多年来引起了广泛关注。我们在一种特定的Baire范畴意义上，对$\nmathbb{R}^d$上几乎所有范数给出了这两个问题的本质紧确答案。对于单位距离问题，我们证明对于$\nmathbb{R}^d$上几乎所有范数$\\|.\\|$，任意$n$个点根据$\\|.\\|$定义的单位距离至多为$\frac{1}{2} d \cdot n \log_2 n$。我们还证明了这一结果本质上是紧确的，因为对于$\nmathbb{R}^d$上的每个范数$\\|.\\|$，对于任意大的$n$，我们都能找到$n$个点，根据$\\|.\\|$定义的单位距离至少为$\frac{1}{2}(d-1-o(1))\cdot n \log_2 n$。对于不同距离问题，我们证明对于$\nmathbb{R}^d$上几乎所有范数$\\|.\\|$，任意$n$个点根据$\\|.\\|$定义的不同距离至少为$(1-o(1))n$。这显然在$o(1)$项以内是紧确的。我们还回答了著名的Hadwiger--Nelson问题对于$\nmathbb{R}^2$上几乎所有范数的情况，表明其单位距离图的色数为$4$。我们的结果以强有力且有些出人意料的形式解决了Brass、Matoušek、Brass-Moser-Pach、Chilakamarri和Robertson的问题和猜想。证明结合了组合与几何思想以及线性代数、拓扑和代数几何的工具。
英文摘要：
Erdős' unit distance problem and Erdős' distinct distances problem are among the most classical and well-known open problems in discrete mathematics. They ask for the maximum number of unit distances, or the minimum number of distinct distances, respectively, determined by $n$ points in the Euclidean plane. The question of what happens in these problems if one considers normed spaces other than the Euclidean plane has been raised in the 1980s by Ulam and Erdős and attracted a lot of attention over the years. We give an essentially tight answer to both questions for almost all norms on $\mathbb{R}^d$, in a certain Baire categoric sense. For the unit distance problem we prove that for almost all norms $\|.\|$ on $\mathbb{R}^d$, any set of $n$ points defines at most $\frac{1}{2} d \cdot n \log_2 n$ unit distances according to $\|.\|$. We also show that this is essentially tight, by proving that for every norm $\|.\|$ on $\mathbb{R}^d$, for any large $n$, we can find $n$ points defining at least $\frac{1}{2}(d-1-o(1))\cdot n \log_2 n$ unit distances according to $\|.\|$. For the distinct distances problem, we prove that for almost all norms $\|.\|$ on $\mathbb{R}^d$ any set of $n$ points defines at least $(1-o(1))n$ distinct distances according to $\|.\|$. This is clearly tight up to the $o(1)$ term. We also answer the famous Hadwiger--Nelson problem for almost all norms on $\mathbb{R}^2$, showing that their unit distance graph has chromatic number $4$. Our results settle, in a strong and somewhat surprising form, problems and conjectures of Brass, Matoušek, Brass-Moser-Pach, Chilakamarri, and Robertson. The proofs combine combinatorial and geometric ideas with tools from Linear Algebra, Topology and Algebraic Geometry.
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