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

We consider the $C^1$ linearization of a perturbed vector field $\omega +P$ over the infinite-dimensional torus ${\mathbb{T}}^{\mathrm{\infty }}$, and determine the sharp regularity requirement of the perturbation $P$ conjugating the unperturbed one $\omega$ onto $\omega -\tilde{\omega }+P$ via a small modifying term $\tilde{\omega }$. We discuss the Diophantine type introduced by Bourgain, investigate the universal nonresonance, and provide the weakest known regularity of perturbations for which KAM applies. Our results allow for lower regularity than analyticity such as Gevrey regularity or even $C^{\mathrm{\infty }}$ regularity. We propose a new KAM scheme with a balancing sequence to overcome the non-polynomial nonresonance, differing from the usual Newtonian approach. Thereby, in addition to deriving the sharp Gevrey exponent along Diophantine nonresonance, we answer the fundamental question of what the minimum regularity required for KAM is in the infinite-dimensional setting: $C^{\mathrm{\infty }}$ regularity. Our linearization technique also applies to the quasi-periodic case over ${\mathbb{T}}^n$.