For a twist map f of the annulus preserving the Lebesgue measure, we give sufficient conditions to assure the existence of a set of positive measure of points with non-zero asymptotic torsion. In particular, we deduce that every bounded instability region for f contains a set of positive measure of points with non-zero asymptotic torsion. Moreover, for an exact symplectic twist map f , we provide a simple, geometric proof of a result by Cheng and Sun (see [CS96]) which characterizes C 0-integrability of f by the absence of conjugate points.