Bourgain's discretization theorem

TL;DR

This paper presents a detailed proof of Bourgain's discretization theorem, improves it for embeddings into Lp spaces, and derives new lower bounds for embeddings of certain metric spaces into L1.

Contribution

It provides a detailed exposition of Bourgain's theorem, offers an improved discretization bound for Lp spaces, and establishes stronger lower bounds for Earthmover metric embeddings into L1.

Findings

Detailed proof of Bourgain's discretization theorem.

Improved discretization bound for embeddings into Lp spaces.

New lower bound of rom or Earthmover metric embeddings into L1.

Abstract

Bourgain's discretization theorem asserts that there exists a universal constant $C\in (0,\infty)$ with the following property. Let $X,Y$ be Banach spaces with $\dim X=n$. Fix $D\in (1,\infty)$ and set $\delta= e^{-n^{Cn}}$. Assume that $\mathcal N$ is a $\delta$-net in the unit ball of $X$ and that $\mathcal N$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $D$. Then the entire space $X$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $CD$. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain's theorem. We also obtain an improvement of Bourgain's theorem in the important case when $Y=L_p$ for some $p\in [1,\infty)$: in this case it suffices to take $\delta= C^{-1}n^{-5/2}$ for the same conclusion to hold true. The case $p=1$ of this improved discretization result has the following consequence. For arbitrarily large $n\in \mathbb{N}$ there exists a family $\mathscr Y$ of $n$-point subsets of ${1,...,n}^2\subseteq \mathbb{R}^2$ such that if we write $|\mathscr Y|= N$ then any $L_1$ embedding of $\mathscr Y$, equipped with the Earthmover metric (a.k.a. transportation cost metric or minimumum weight matching metric) incurs distortion at least a constant multiple of $\sqrt{\log\log N}$; the previously best known lower bound for this problem was a constant multiple of $\sqrt{\log\log \log N}$.