Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on ${\mathbb{Z}}^d$

We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on ${\mathbb{Z}}^d$, thus extending Anderson localization from the linear (cf. Bourgain [Geom. Funct. Anal., 17(3):682–706, 2007]) to a nonlinear setting, and the random (cf. Bourgain-Wang [J. Eur. Math. Soc., 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 [Geom. Funct. Anal., 17(3):682–706, 2007].