Assouad's theorem with dimension independent of the snowflaking

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, themetric 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.