{"title":"典型规范中的单位和不同距离","authors":"Noga Alon, Matija Bucić, Lisa Sauermann","doi":"10.1007/s00039-025-00698-x","DOIUrl":null,"url":null,"abstract":"<p>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 <i>n</i> 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 <span>\\(\\mathbb{R}^{d}\\)</span>, in a certain Baire categoric sense.</p><p>For the unit distance problem we prove that for almost all norms ∥.∥ on <span>\\(\\mathbb{R}^{d}\\)</span>, any set of <i>n</i> points defines at most <span>\\(\\frac{1}{2} d \\cdot n \\log _{2} n\\)</span> unit distances according to ∥.∥. We also show that this is essentially tight, by proving that for <i>every</i> norm ∥.∥ on <span>\\(\\mathbb{R}^{d}\\)</span>, for any large <i>n</i>, we can find <i>n</i> points defining at least <span>\\(\\frac{1}{2}(d-1-o(1))\\cdot n \\log _{2} n\\)</span> unit distances according to ∥.∥.</p><p>For the distinct distances problem, we prove that for almost all norms ∥.∥ on <span>\\(\\mathbb{R}^{d}\\)</span> any set of <i>n</i> points defines at least (1−<i>o</i>(1))<i>n</i> distinct distances according to ∥.∥. This is clearly tight up to the <i>o</i>(1) term.</p><p>We also answer the famous Hadwiger–Nelson problem for almost all norms on <span>\\(\\mathbb{R}^{2}\\)</span>, showing that their unit distance graph has chromatic number 4.</p><p>Our results settle, in a strong and somewhat surprising form, problems and conjectures of Brass, Matoušek, Brass–Moser–Pach, Chilakamarri, and Robertson. 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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 <span>\\\\(\\\\mathbb{R}^{d}\\\\)</span>, in a certain Baire categoric sense.</p><p>For the unit distance problem we prove that for almost all norms ∥.∥ on <span>\\\\(\\\\mathbb{R}^{d}\\\\)</span>, any set of <i>n</i> points defines at most <span>\\\\(\\\\frac{1}{2} d \\\\cdot n \\\\log _{2} n\\\\)</span> unit distances according to ∥.∥. We also show that this is essentially tight, by proving that for <i>every</i> norm ∥.∥ on <span>\\\\(\\\\mathbb{R}^{d}\\\\)</span>, for any large <i>n</i>, we can find <i>n</i> points defining at least <span>\\\\(\\\\frac{1}{2}(d-1-o(1))\\\\cdot n \\\\log _{2} n\\\\)</span> unit distances according to ∥.∥.</p><p>For the distinct distances problem, we prove that for almost all norms ∥.∥ on <span>\\\\(\\\\mathbb{R}^{d}\\\\)</span> any set of <i>n</i> points defines at least (1−<i>o</i>(1))<i>n</i> distinct distances according to ∥.∥. This is clearly tight up to the <i>o</i>(1) term.</p><p>We also answer the famous Hadwiger–Nelson problem for almost all norms on <span>\\\\(\\\\mathbb{R}^{2}\\\\)</span>, showing that their unit distance graph has chromatic number 4.</p><p>Our results settle, in a strong and somewhat surprising form, problems and conjectures of Brass, Matoušek, Brass–Moser–Pach, Chilakamarri, and Robertson. 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引用次数: 0
摘要
Erdős“单位距离问题”和Erdős“不同距离问题”是离散数学中最经典和最著名的开放问题。它们要求单位距离的最大值,或不同距离的最小值,分别由欧几里得平面上的n个点决定。如果考虑欧几里得平面以外的赋范空间,在这些问题中会发生什么?这个问题在20世纪80年代由Ulam和Erdős提出,多年来引起了很多关注。在一定的贝尔范畴意义上,我们对\(\mathbb{R}^{d}\)上几乎所有的规范给出了一个本质上严密的答案。对于单位距离问题,我们证明了对于几乎所有规范∥。∥在\(\mathbb{R}^{d}\)上,任意n个点的集合根据∥.∥定义最多\(\frac{1}{2} d \cdot n \log _{2} n\)个单位距离。我们也证明了这本质上是紧密的,通过证明对于每一个范数∥。∥在\(\mathbb{R}^{d}\)上,对于任意大的n,我们可以根据∥.∥找到n个定义至少\(\frac{1}{2}(d-1-o(1))\cdot n \log _{2} n\)单位距离的点。对于明显距离问题,我们证明了对于几乎所有规范∥。∥在\(\mathbb{R}^{d}\)上任意n个点的集合根据∥.∥定义了至少(1−o(1))n个不同的距离。这很明显是紧到0(1)项。我们还对\(\mathbb{R}^{2}\)上几乎所有的范数回答了著名的Hadwiger-Nelson问题,证明了它们的单位距离图的色数为4。我们的结果解决了Brass, Matoušek, Brass - moser - pach, Chilakamarri和Robertson的问题和猜想。证明将组合和几何思想与线性代数、拓扑和代数几何的工具结合起来。
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.
期刊介绍:
Geometric And Functional Analysis (GAFA) publishes original research papers of the highest quality on a broad range of mathematical topics related to geometry and analysis.
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Publishes major results on topics in geometry and analysis.
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