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| We present a KAM type theorem for two-dimensional tori which does not assume the classical diophantine condition, instead it is based on finite approximation of the rotation vector. Specifically, we prove that for all $p>0$ a KAM theorem holds with a perturbation threshold $\varepsilon_*$, which depends only on $N(p)$ first "digits" of the continued fraction expansion of the rotation frequency $\omega$ provided that $\omega$ belongs to a set of probabilistic measure $1-p$. The quantities are explicitly computed, which makes the method amenable for computer assisted proofs. Our reasoning relies on a careful analysis of the small divisors - the ones which are "truly small" are related to the continued fraction expansion of $\omega$, which can be controlled statistically in spirit of the theorem on Khintchine's constant. |
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