Yet another short proof of Bourgain's distorsion estimate

TL;DR

This paper presents a concise, elementary proof of Bourgain's theorem that regular trees cannot be embedded into uniformly convex Banach spaces with a bi-Lipschitz map, using a self-improvement argument.

Contribution

It introduces a simplified proof technique for Bourgain's distortion estimate, making the result more accessible and easier to understand.

Findings

Regular trees do not admit bi-Lipschitz embeddings into uniformly convex Banach spaces.

The proof is shorter and more elementary than previous approaches.

A self-improvement argument is used to establish the result.

Abstract

We use a self-improvement argument to give a very short and elementary proof of the result of Bourgain saying that regular trees do not admit bi-Lipschitz embeddings into uniformly convex Banach spaces.