  It is well-known that density estimation on the unit interval is
asymptotically equivalent to a Gaussian white noise experiment, provided the
densities have H\"older smoothness larger than $1/2$ and are uniformly bounded
away from zero. We derive matching lower and constructive upper bounds for the
Le Cam deficiencies between these experiments, with explicit dependence on both
the sample size and the size of the densities in the parameter space. As a
consequence, we derive sharp conditions on how small the densities can be for
asymptotic equivalence to hold. The related case of Poisson intensity
estimation is also treated.
