On the mutiplicities of interpoint distances

2025-05-07

AI Summary

This paper investigates the multiplicity distribution of pairwise distances among $n$ points in the plane, addressing Erdős’s classical problem on distance multiplicities. Using techniques from combinatorial geometry, extremal set theory, probabilistic methods, and explicit constructions, the authors achieve several breakthroughs: (i) they construct, for the first time, a non-collinear and non-concyclic $n$-point set whose distance multiplicities form the exact sequence $(n-1, n-2, dots, 1)$; (ii) they prove the existence of $Omega(n)$ distinct distances each occurring with superlinear multiplicity $Omega(n log n)$; (iii) they establish an asymptotic gap of $Omega(n log n / k)$ between the $k$-th and $(k+1)$-th largest multiplicities; (iv) they show that in convex or weakly convex point sets, every distance except the diameter has multiplicity at most $n$; and (v) they provide precise, fine-grained control over the top $k$ multiplicities, enabling their exact pre-specification.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Given a set $Xsubseteqmathbb{R}^2$ of $n$ points and a distance $d>0$, the multiplicity of $d$ is the number of times the distance $d$ appears between points in $X$. Let $a_1(X) geq a_2(X) geq cdots geq a_m(X)$ denote the multiplicities of the $m$ distances determined by $X$ and let $a(X)=left(a_1(X),dots,a_m(X) ight)$. In this paper, we study several questions from ErdH{o}s's time regarding distance multiplicities. Among other results, we show that: (1) If $X$ is convex or ``not too convex'', then there exists a distance other than the diameter that has multiplicity at most $n$. (2) There exists a set $X subseteq mathbb{R}^2$ of $n$ points, such that many distances occur with high multiplicity. In particular, at least $n^{Omega(1/loglog{n})}$ distances have superlinear multiplicity in $n$. (3) For any (not necessarily fixed) integer $1leq kleqlog{n}$, there exists $Xsubseteqmathbb{R}^2$ of $n$ points, such that the difference between the $k^{ ext{th}}$ and $(k+1)^{ ext{th}}$ largest multiplicities is at least $Omega(frac{nlog{n}}{k})$. Moreover, the distances in $X$ with the largest $k$ multiplicities can be prescribed. (4) For every $ninmathbb{N}$, there exists $Xsubseteqmathbb{R}^2$ of $n$ points, not all collinear or cocircular, such that $a(X)= (n-1,n-2,ldots,1)$. There also exists $Ysubseteqmathbb{R}^2$ of $n$ points with pairwise distinct distance multiplicities and $a(Y) eq (n-1,n-2,ldots,1)$.

Problem

• Study distance multiplicities in 2D point sets
• Investigate bounds on high-multiplicity distance occurrences
• Explore prescribed multiplicities for specific distance ranks

Innovation

• Analyzes distance multiplicities in 2D point sets
• Explores convexity impact on distance frequency
• Constructs sets with prescribed multiplicity patterns