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到希尔伯特空间的 $n$ 点度量空间，那么对于每个 $\tau>0$，都存在 $\mathcal{M}$ 的一个随机子集 $\mathcal{Z}$，使得对于任意满足 $d(x,y)\ge \tau$ 的点对 $x,y\in \mathcal{M}$，$x\in \mathcal{Z}$ 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 $\kappa>1$ 是一个普适常数，而 $\beta>0$ 仅依赖于拟对称嵌入的模。该证明依赖于对 Arora--Rao--Vazirani 舍入技术的改进。该结果的应用包括：$\ell_1$ 中 $n$ 点子集可能的最大欧几里得畸变为 $\Theta(\sqrt{\log n})$，以及针对规模为 $n$ 的输入的稀疏割问题的 Goemans--Linial 半定规划的整数性间隙为 $\Theta(\sqrt{\log n})$。文中还给出了若干进一步的应用。