标题： 关于点间距离的重数 显示英文标题

标题： On multiplicities of interpoint distances

摘要： 给定一个包含$n$个点的集合$X\subseteq\mathbb{R}^2$和一个距离$d>0$，$d$的重数是指距离$d$在$X$中的点之间出现的次数。 设$a_1(X) \geq a_2(X) \geq \cdots \geq a_m(X)$表示由$X$确定的$m$距离的重数，并让$a(X)=\left(a_1(X),\dots,a_m(X)\right)$。 在本文中，我们研究了Erdős时期关于距离重数的几个问题。 在其他结果中，我们证明了： (1) 如果$X$是凸的或“不太凸”，那么存在一个不同于直径的距离，其重数至多为$n$。 (2) 存在一个包含$n$个点的集合$X \subseteq \mathbb{R}^2$，使得许多距离出现的次数很高。 特别地，至少$n^{\Omega(1/\log\log{n})}$个距离在$n$中具有超线性次数。 (3) 对于任何（不一定固定的）整数$1\leq k\leq\log{n}$，存在$X\subseteq\mathbb{R}^2$的$n$个点，使得$k^{\text{th}}$和$(k+1)^{\text{th}}$最大的重数之间的差至少为$\Omega(\frac{n\log{n}}{k})$。 此外，具有最大$k$重数的$X$中的距离可以被指定。(4) 对于每个$n\in\mathbb{N}$，存在$X\subseteq\mathbb{R}^2$的$n$个点，不全共线或共圆，使得$a(X)= (n-1,n-2,\ldots,1)$。 也存在具有成对不同距离多重性的 $Y\subseteq\mathbb{R}^2$ 个 $n$ 点以及 $a(Y) \neq (n-1,n-2,\ldots,1)$。 显示英文摘要

摘要： 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 Erd\H{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 $n^{\Omega(1/\log\log{n})}$ 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 $\Omega(\frac{n\log{n}}{k})$. 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 $a(Y) \neq (n-1,n-2,\ldots,1)$.