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We consider linearization of perturbed vector field $ \omega+P $ overinfinite dimensional torus $ \mathbb{T}^\infty $ and give sharp regularityrequirement for perturbation $ P $ under which there is a nearly identicaltransformation conjugating the unperturbed one $ \omega $ onto $\omega-\tilde{\omega}+P $ via a small modifying term $ \tilde{\omega} $.Besides discussing the Diophantine type introduced by Bourgain [11], we alsoinvestigate the universal nonresonance and provide weakest regularity ofperturbations known so far for which KAM applies. Lower than analyticity, ourresults allow Gevrey or even only $ C^\infty $, and the new KAM scheme with abalancing sequence to overcome non-polynomial nonresonance is shown to benon-Newtonian that differs from the usual ones. Thereby, except deriving sharpGevrey exponent along Diophantine nonresonance, we answer the fundamentalquestion of what is the minimum regularity required for KAM in infinitedimensional case. Additionally, our linearization could also be employed todeal with quasi periodic case over $ \mathbb{T}^n $.

Towards sharp regularity: Full dimensional tori in $ C^\infty $ vector fields over $ \mathbb{T}^\infty $