Assouad’s theorem with dimension independent of the snowflaking

Abstract

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