Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data

We study the cubic defocusing nonlinear Schrödinger equation on R4 with supercritical initial data. For randomized data in Hs(R4), we prove almost sure local wellposedness for 17<s<1 and scattering 57<s<1. The randomization is based a unit-scale decomposition frequency space, angular variable, – result an additional physical space. employ new probabilistic estimates linear flow data, where effectively combine advantages of different decompositions.