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 $a(Y) \neq (n-1,n-2,\ldots,1)$.

翻译：给定平面点集 $X\subseteq\mathbb{R}^2$ 包含 $n$ 个点以及距离 $d>0$，距离 $d$ 的重数定义为 $X$ 中点对之间出现该距离的次数。令 $a_1(X) \geq a_2(X) \geq \cdots \geq a_m(X)$ 表示 $X$ 所确定的 $m$ 个距离的重数，并记 $a(X)=\left(a_1(X),\dots,a_m(X)\right)$。本文研究了 Erdős 提出的若干关于距离重数的问题。主要结果包括：(1) 若 $X$ 为凸集或“非强凸集”，则存在除直径外的某个距离，其重数不超过 $n$。(2) 存在由 $n$ 个点构成的平面点集 $X \subseteq \mathbb{R}^2$，使得大量距离具有高重数。具体而言，至少存在 $n^{Ω(1/\log\log{n})}$ 个距离具有关于 $n$ 的超线性重数。(3) 对任意（不必固定）整数 $1\leq k\leq\log{n}$，存在 $n$ 点平面点集 $X\subseteq\mathbb{R}^2$，使得第 $k$ 大与第 $(k+1)$ 大重数之差至少为 $Ω(\frac{n\log{n}}{k})$。此外，$X$ 中具有前 $k$ 大重数的距离可被预先指定。(4) 对任意 $n\in\mathbb{N}$，存在非共线且非共圆的 $n$ 点平面点集 $X\subseteq\mathbb{R}^2$，满足 $a(X)= (n-1,n-2,\ldots,1)$。同时存在 $n$ 点平面点集 $Y\subseteq\mathbb{R}^2$，其距离重数两两不同且 $a(Y) \neq (n-1,n-2,\ldots,1)$。