  We prove global well-posedness and scattering for the defocusing, cubic NLS
on $\mathbb{R}^3$ with initial data in $H^s(\mathbb{R}^3)$ for $s>49/74$. The
proof combines the ideas of resonance decomposition in \cite{CKSTT4} and
linear-nonlinear decomposition in \cite{ben1}\cite{roy} together with the idea
of large time iteration.
