On multiplicities of interpoint distances

TL;DR

This work investigates how often interpoint distances appear among $n$ points in the plane by studying the multiplicities $a_k(X)$ and the ordered vector $a(X)$. It advances Erdős's distance-program through four main directions: (i) establishing that a second distance with multiplicity at most $n$ occurs in convex or not-too-convex configurations; (ii) constructing $n$-point sets with many distances having superlinear multiplicity, including $n^{\Omega(1/\log\log n)}$ such distances; (iii) proving lower bounds on gaps $a_k(X)-a_{k+1}(X)$ with the possibility to prescribe the top-$k$ multiplicities; and (iv) producing examples where $a(X)=(n-1,n-2,\dots,1)$ or where all multiplicities are distinct. The results blend convex-hull layer arguments, grid-based constructions, and sum-of-two-squares techniques to illuminate the distribution of distance multiplicities and connect to the broader unit-distance problem in combinatorial geometry.

Abstract

Given a set $X\subseteq\mathbb{R}^2$ 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 $a(X)=\left(a_1(X),\dots,a_m(X)\right)$. In this paper, we study several questions from Erdős'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 $n^{Ω(1/\log\log{n})}$ distances have superlinear multiplicity in $n$. (3) For any (not necessarily fixed) integer $1\leq k\leq\log{n}$, there exists $X\subseteq\mathbb{R}^2$ of $n$ points, such that the difference between the $k^{\text{th}}$ and $(k+1)^{\text{th}}$ largest multiplicities is at least $Ω(\frac{n\log{n}}{k})$. Moreover, the distances in $X$ with the largest $k$ multiplicities can be prescribed. (4) For every $n\in\mathbb{N}$, there exists $X\subseteq\mathbb{R}^2$ of $n$ points, not all collinear or cocircular, such that $a(X)= (n-1,n-2,\ldots,1)$. There also exists $Y\subseteq\mathbb{R}^2$ of $n$ points with pairwise distinct distance multiplicities and $a(Y) \neq (n-1,n-2,\ldots,1)$.

Figures (4)

• Figure 1: The construction in Proposition \ref{['thm:cons']}, when $m=7$. Dotted edges represent the second largest distance $\Delta_2$.
• Figure 2: Multiplicities of distances in the grid.
• Figure 3: Three configurations $X$ satisfying $a(X)=(n-1,n-2,\ldots,1)$ for $n=7$.
• Figure 4: Point sets ($n=9$ and $n=10$) with pairwise distinct distance multiplicities and $a(X) \neq (n-1,n-2,\ldots,1)$.

Theorems & Definitions (22)

• Conjecture 1.1: Erdős Er84, see also EF95a
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• proof
• Proposition 1.5
• Theorem 1.7
• Proposition 1.8
• Theorem 1.9
• Corollary 1.10