  In this paper we address the question of the pointwise almost everywhere
limit of nonlinear Schr\"odinger flows to the initial data, in both the
continuous and the periodic settings. Then we show how, in some cases, certain
smoothing effects for the non-homogeneous part of the solution can be used to
upgrade to a uniform convergence to zero of this part, and we discuss the
sharpness of the results obtained. We also use randomization techniques to
prove that with much less regularity of the initial data, both in continuous
and the periodic settings, almost surely one obtains uniform convergence of the
nonlinear solution to the initial data, hence showing how more generic results
can be obtained.
