  We consider the cubic nonlinear Schr\"odinger equation on $2$-dimensional
irrational tori. We construct solutions which undergo growth of Sobolev norms.
More concretely, for every $s>0$, $s\neq 1$ and almost every choice of spatial
periods we construct solutions whose $H^s$ Sobolev norms grow by any prescribed
factor. Moreover, for a set of spatial periods with positive Hausdorff
dimension we construct solutions whose Sobolev norms go from arbitrarily small
to arbitrarily large. We also provide estimates for the time needed to undergo
the norm explosion. Note that the irrationality of the space periods decouples
the linear resonant interactions into products of $1$-dimensional resonances,
reducing considerably the complexity of the resonant dynamics usually used to
construct transfer of energy solutions. However, one can provide these growth
of Sobolev norms solutions by using quasi-resonances relying on Diophantine
approximation properties of the space periods.
