Standard time-stepping techniques require a regularity constraint on the initial data u₀ for dispersive equations. We introduce a class of filtered integrators for problems where certain constraints are not satisfied. Moreover, when the regularity is critically low (u₀∈H^s with s≤d/2),the classical stability argument based on Sobolev spaces does not hold. We have developed a framework of Bourgain spaces that overcomes this problem. In this talk, I will use the example of cubic nonlinear Schrodinger equation to summarize how these techniques are applied.