Relation between irrationality and regularity for $C^1$ conjugacy of $C^2$ circle diffeomorphisms to rigid rotations

By introducing the modulus of continuity, we first establish the corresponding cross-ratio distortion estimates under $C^2$ smoothness, and further derive a Denjoy-type inequality, which is almost optimal for dealing with circle diffeomorphisms. The latter plays a prominent role in the study of $C^1$ conjugacy to irrational rotations. We also establish an explicit integrability correlation between continuity and irrationality for the first time. Furthermore, the regularity of the conjugation is addressed and proved to be sharp.