  We study linear and non-linear small divisors problems in analytic and
non-analytic regularity. We observe that the Bruno arithmetic condition, which
is usually attached to non-linear analytic problems, can also be characterized
as the optimal condition to solve the linear problem in some fixed non
quasi-analytic class. Based on this observation, it is natural to conjecture
that the optimal arithmetic condition for the linear problem is also optimal
for non-linear small divisors problems in any reasonable non quasi-analytic
classes. Our main result proves this conjecture in a representative non-linear
problem, which is the linearization of vector fields on the torus, in the most
representative non quasi-analytic class, which is the Gevrey class. The proof
follows Moser's argument of approximation by analytic functions, and uses in an
essential way works of Popov, R\"{u}ssmann and P\"{o}schel.
