Metric dimension reduction modulus for superlogarithmic distortion

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.