  In this paper we generalize Nesterenko's criterion to the case where the
small linear forms have an oscillating behaviour (for instance given by the
saddle point method). This criterion provides both a lower bound for the
dimension of the vector space spanned over the rationals by a family of real
numbers, and a measure of simultaneous approximation to these numbers (namely,
an upper bound for the irrationality exponent if 1 and only one other number
are involved). As an application, we prove an explicit measure of simultaneous
approximation to $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, and $\zeta(11)$, using
Zudilin's proof that at least one of these numbers is irrational.
