  In this paper, we propose a new scheme for the integration of the periodic
nonlinear Schr\"{o}dinger equation and rigorously prove convergence rates at
low regularity. The new integrator has decisive advantages over standard
schemes at low regularity. In particular, it is able to handle initial data in
$H^s$ for $0 < s\le 1$. The key feature of the integrator is its ability to
distinguish between low and medium frequencies in the solution and to treat
them differently in the discretization. This new approach requires a
well-balanced filtering procedure which is carried out in Fourier space. The
convergence analysis of the proposed scheme is based on discrete (in time)
Bourgain space estimates which we introduce in this paper. A numerical
experiment illustrates the superiority of the new integrator over standard
schemes for rough initial data.
