We prove that if $(\mathcal{M},d)$ is an $n$-point metric space that embeds quasisymmetrically into a Hilbert space, then for every $\tau>0$ there is a random subset $\mathcal{Z}$ of $\mathcal{M}$ such that for any pair of points $x,y\in \mathcal{M}$ with $d(x,y)\ge \tau$, the probability that both $x\in \mathcal{Z}$ and $d(y,\mathcal{Z})\ge \beta\tau/\sqrt{1+\log (|B(y,\kappa \beta \tau)|/|B(y,\beta \tau)|)}$ is $\Omega(1)$, where $\kappa>1$ is a universal constant and $\beta>0$ depends only on the modulus of the quasisymmetric embedding. The proof relies on a refinement of the Arora--Rao--Vazirani rounding technique. Among the applications of this result is that the largest possible Euclidean distortion of an $n$-point subset of $\ell_1$ is $\Theta(\sqrt{\log n})$, and the integrality gap of the Goemans--Linial semidefinite program for the Sparsest Cut problem on inputs of size $n$ is $\Theta(\sqrt{\log n})$. Multiple further applications are given.

翻译：我们证明，如果$(\mathcal{M},d)$是一个能拟对称嵌入希尔伯特空间的$n$点度量空间，则对于任意$\tau>0$，存在$\mathcal{M}$的随机子集$\mathcal{Z}$，使得对于任意满足$d(x,y)\ge \tau$的点对$x,y\in \mathcal{M}$，同时满足$x\in \mathcal{Z}$且$d(y,\mathcal{Z})\ge \beta\tau/\sqrt{1+\log (|B(y,\kappa \beta \tau)|/|B(y,\beta \tau)|)}$的概率为$\Omega(1)$，其中$\kappa>1$为普适常数，$\beta>0$仅取决于拟对称嵌入的模。该证明基于对Arora--Rao--Vazirani舍入技术的改进。此结果的应用包括：$\ell_1$中$n$点子集可能达到的最大欧几里得畸变为$\Theta(\sqrt{\log n})$，以及Goemans--Linial半定规划在规模为$n$的稀疏割问题输入上的整数性间隙为$\Theta(\sqrt{\log n})$。本文还给出了若干其他应用。