# Assouad’s theorem with dimension independent of the snowflaking

### Assaf Naor

New York University, United States### Ofer Neiman

Ben Gurion University of the Negev, Beer Sheva, Israel

## Abstract

It is shown that for every $K>0$ and $ε∈(0,1/2)$ there exist $N=N(K)∈N$ and $D=D(K,ε)∈(1,∞)$ with the following properties. For every metric space $(X,d)$ with doubling constant at most $K$, the metric space $(X,d_{1−ε})$ admits a bi-Lipschitz embedding into $R_{N}$ with distortion at most $D$. The classical Assouad embedding theorem makes the same assertion, but with $N→∞$ as $ε→0$.

## Cite this article

Assaf Naor, Ofer Neiman, Assouad’s theorem with dimension independent of the snowflaking. Rev. Mat. Iberoam. 28 (2012), no. 4, pp. 1123–1142

DOI 10.4171/RMI/706