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Using a Birkhoff normal form transform to impede mode transfer in a finite"barrier", we prove localization of arbitrary $\ell^2$ data for polynomiallylong time for the nonlinear quasi-periodic Schr\"odinger equation on $\mathbbZ^d$.

Long-time Anderson Localization for the Nonlinear quasi-periodic Schr\"odinger Equation on $\mathbb Z^d$