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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 everyseparable metric space $(X,d)$ with doubling constant at most $K$, the metricspace $(X,d^{1-\e})$ admits a bi-Lipschitz embedding into $\R^N$ withdistortion at most $D$. The classical Assouad embedding theorem makes the sameassertion, but with $N\to \infty$ as $\e\to 0$.

Assouad's theorem with dimension independent of the snowflaking