Metric dimension reduction modulus for superlogarithmic distortion

Abstract

The metric dimension reduction modulus $k^α_n(\ell_\infty)$ is the smallest $k$ such that every $n$--point metric space can be embedded into some $k$-dimensional normed space, with bi--Lipschitz distortion at most $α$. Determining sharp asymptotics for $k^α_n(\ell_\infty)$ is a fundamental task in metric geometry, with $α=Θ(\log n)$ bearing particular interest. A line of advances over the past decades has led to an upper bound on $k^α_n(\ell_\infty)$ for $α= Ω(\log n)$, but a matching lower bound has remained open. We close this gap, establishing: for every fixed $β> 0$, $$ k^α_n(\ell_\infty) =Θ\bigg(\frac{\log n}{\log(\fracα{\log n}+1)}\bigg)\quad \mbox{for every $α\geq β\log n$}. $$ This resolves a question from Naor's 2018 ICM plenary lecture. Our result is obtained by characterizing the minimum dimension $d$ for which, with high probability, a random regular graph admits an $α$--embedding into some $d$--dimensional normed space.