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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_698_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{R}^{d}$</EquationSource> </InlineEquation>, in a certain Baire categoric sense.For the unit distance problem we prove that for almost all norms ∥.∥ on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_698_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{R}^{d}$</EquationSource> </InlineEquation>, any set of n points defines at most <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_698_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{1}{2} d \cdot n \log _{2} n$</EquationSource> </InlineEquation> unit distances according to ∥.∥. We also show that this is essentially tight, by proving that for every norm ∥.∥ on <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_698_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{R}^{d}$</EquationSource> </InlineEquation>, for any large n, we can find n points defining at least <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_698_Article_IEq5.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="175" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{1}{2}(d-1-o(1))\cdot n \log _{2} n$</EquationSource> </InlineEquation> unit distances according to ∥.∥.For the distinct distances problem, we prove that for almost all norms ∥.∥ on <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_698_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{R}^{d}$</EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_698_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{R}^{2}$</EquationSource> </InlineEquation>, 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.

Unit and Distinct Distances in Typical Norms

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发表日期：

2025年1月24日

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Noga Alon,
Matija Bucić,
Lisa Sauermann

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.