  A fundamental question in Dynamical Systems is to identify regions of
phase/parameter space satisfying a given property (stability, linearization,
etc). Given a family of analytic circle diffeomorphisms depending on a
parameter, we obtain effective (almost optimal) lower bounds of the Lebesgue
measure of the set of parameters that are conjugated to a rigid rotation. We
estimate this measure using an a-posteriori KAM scheme that relies on
quantitative conditions that are checkable using computer-assistance. We
carefully describe how the hypotheses in our theorems are reduced to a finite
number of computations, and apply our methodology to the case of the Arnold
family. Hence we show that obtaining non-asymptotic lower bounds for the
applicability of KAM theorems is a feasible task provided one has an
a-posteriori theorem to characterize the problem. Finally, as a direct
corollary, we produce explicit asymptotic estimates in the so called local
reduction setting (\`a la Arnold) which are valid for a global set of
rotations.
