  We consider the cubic nonlinear Schr\"odinger equation posed on the spatial
domain $\mathbb{R}\times \mathbb{T}^d$. We prove modified scattering and
construct modified wave operators for small initial and final data respectively
($1\leq d\leq 4)$. The key novelty comes from the fact that the modified
asymptotic dynamics are dictated by the resonant system of this equation, which
sustains interesting dynamics when $d\geq 2$. As a consequence, we obtain
global solutions to the defocusing and focusing problems on $\mathbb{R}\times
\mathbb{T}^d$ (for any $d\geq 2$) with infinitely growing high Sobolev norms
$H^s$.
