  We consider testing for presence of a signal in Gaussian white noise with
intensity 1/sqrt(n), when the alternatives are given by smoothness ellipsoids
with an L2-ball of (squared) radius rho removed. It is known that, for a fixed
Sobolev type ellipsoid of smoothness beta and size M, a rho which is of order n
to the power -4 beta/(4 beta+1)} is the critical separation rate, in the sense
that the minimax error of second kind over alpha-tests stays asymptotically
between 0 and 1 strictly (Ingster, 1982). In addition, Ermakov (1990) found the
sharp asymptotics of the minimax error of second kind at the separation rate.
For adaptation over both beta and M in that context, it is known that a
loglog-penalty over the separation rate for rho is necessary for a nonzero
asymptotic power. Here, following an example in nonparametric estimation
related to the Pinsker constant, we investigate the adaptation problem over the
ellipsoid size M only, for fixed smoothness degree beta. It is established that
the sharp risk asymptotics can be replicated in that adaptive setting, if rho
tends to zero slower than the separation rate. The penalty for adaptation here
turns out to be a sequence tending to infinity arbitrarily slowly.
