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We study multi-frequency quasi-periodic Schr\"odinger operators on$\mathbb{Z}$ in the regime of positive Lyapunov exponent and for generalanalytic potentials. Combining Bourgain's semi-algebraic elimination ofmultiple resonances with the method of elimination of double resonances viaresultants, we establish exponential finite-volume localization as well as theseparation between the eigenvalues. In a follow-up paper we develop the methodfurther to show that for potentials given by large generic trigonometricpolynomials the spectrum consists of a single interval, as conjectured byChulaevski and Sinai.

On localization and the spectrum of multi-frequency quasi-periodic operators