Almost sure well-posedness of the cubic nonlinear Schr\"odinger equation below L^2(T)

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.