Anderson localized states for the quasi-periodic nonlinear Schr\"odinger equation on Zᵈ

We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schr\"odinger equation on Zᵈ, thus extending Anderson localization from the linear (cf. Bourgain [GAFA 17(3), 682--706, 2007]) to a nonlinear setting, and the random (cf. Bourgain-Wang [JEMS 10(1), 1--45, 2008]) to a deterministic setting. Among the main ingredients are a new Diophantine estimate of quasi-periodic functions in arbitrarily dimensional phase space, and the application of Bourgain's geometric lemma in [GAFA 17(3), 682--706, 2007].