  We consider the Cauchy problem for the one-dimensional periodic cubic
nonlinear Schr\"odinger equation (NLS) with initial data below L^2. In
particular, we exhibit nonlinear smoothing when the initial data are
randomized. Then, we prove local well-posedness of NLS almost surely for the
initial data in the support of the canonical Gaussian measures on H^s(T) for
each s > -1/3, and global well-posedness for each s > -1/12.
