  We give a constructive proof of a theorem of Naor and Neiman, (to appear,
Revista Matematica Iberoamercana), which asserts that if $(E,d)$ is a doubling
metric space, there is an integer $N > 0$, that depends only on the metric
doubling constant, such that for each exponent $\alpha \in (1/2,1)$, we can
find a bilipschitz mapping $F = (E,d^{\alpha}) \to \R^N$.
