  We consider the cubic Schr\"odinger equation posed on product spaces subject
to a generic Diophantine condition. Our analysis shows that the small-amplitude
solutions undergo modified scattering to an effective dynamics governed by some
interactions that do not amplify the Sobolev norms. This is in sharp contrast
with the infinite energy cascade scenario observed by
Hani--Pausader--Tzvetkov--Visciglia in the absence of Diophantine conditions.
