Random zero sets with local growth guarantees

Abstract

We prove that if $(\mathcal{M},d)$ is an $n$-point metric space that embeds quasisymmetrically into a Hilbert space, then for every $τ>0$ there is a random subset $\mathcal{Z}$ of $\mathcal{M}$ such that for any pair of points $x,y\in \mathcal{M}$ with $d(x,y)\ge τ$, the probability that both $x\in \mathcal{Z}$ and $d(y,\mathcal{Z})\ge βτ/\sqrt{1+\log (|B(y,κβτ)|/|B(y,βτ)|)}$ is $Ω(1)$, where $κ>1$ is a universal constant and $β>0$ 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 $Θ(\sqrt{\log n})$, and the integrality gap of the Goemans--Linial semidefinite program for the Sparsest Cut problem on inputs of size $n$ is $Θ(\sqrt{\log n})$. Multiple further applications are given.