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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.
Shi et al. (Mon,) studied this question.