Assouad's theorem with dimension independent of the snowflaking

TL;DR

The paper proves that for any doubling metric space, a snowflaked version can be embedded into a finite-dimensional Euclidean space with uniform bounds, independent of the snowflaking parameter.

Contribution

It establishes a dimension-independent version of Assouad's theorem for snowflaked metric spaces, unlike the classical result where dimension grows as snowflaking diminishes.

Findings

Finite-dimensional bi-Lipschitz embeddings for snowflaked spaces

Dimension bounds depend only on doubling constant and parameters

Embedding distortion is uniformly bounded

Abstract

It is shown that for every $K>0$ and $\e\in (0,1/2)$ there exist $N=N(K)\in \N$ and $D=D(K,\e)\in (1,\infty)$ with the following properties. For every separable metric space $(X,d)$ with doubling constant at most $K$, the metric space $(X,d^{1-\e})$ admits a bi-Lipschitz embedding into $\R^N$ with distortion at most $D$. The classical Assouad embedding theorem makes the same assertion, but with $N\to \infty$ as $\e\to 0$.