Metric dimension reduction modulus for superlogarithmic distortion

TL;DR

This paper determines the asymptotic behavior of the metric dimension reduction modulus for superlogarithmic distortion, resolving a longstanding open problem in metric geometry.

Contribution

It establishes the exact asymptotics of the metric dimension reduction modulus for \\ell_\\infty, closing a gap in understanding for \\alpha \\geq \\beta \\log n, and characterizes the minimal embedding dimension via random regular graphs.

Findings

Exact asymptotics for the metric dimension reduction modulus for \\alpha \\geq \\beta \\log n.

Resolution of a question from Naor's 2018 ICM lecture.

Connection between embeddings and properties of random regular graphs.

Abstract

The metric dimension reduction modulus $k^\alpha_n(\ell_\infty)$ 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 $\alpha$. Determining sharp asymptotics for $k^\alpha_n(\ell_\infty)$ is a fundamental task in metric geometry, with $\alpha=\Theta(\log n)$ bearing particular interest. A line of advances over the past decades has led to an upper bound on $k^{\alpha}_n(\ell_\infty)$ for $\alpha = \Omega(\log n)$, but a matching lower bound has remained open. We close this gap, establishing: for every fixed $\beta > 0$, $$ k^{\alpha}_n(\ell_\infty) =\Theta\bigg(\frac{\log n}{\log(\frac{\alpha}{\log n}+1)}\bigg)\quad \mbox{for every $\alpha\geq \beta \log n$}. $$ 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 $\alpha$--embedding into some $d$--dimensional normed space.