# 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 $\varepsilon\in (0,1/2)$ there exist $N=N(K)\in \mathbb{N}$ and $D=D(K,\varepsilon)\in (1,\infty)$ with the following properties. For every metric space $(X,d)$ with doubling constant at most $K$, the metric space $(X,d^{1-\varepsilon})$ 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 $\varepsilon\to 0$.

## 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