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

Sobolev norms explosion for the cubic NLS on irrational tori