We consider a model for a polymer chain interacting with a sequence of equispaced flat interfaces through a pinning potential. The intensity d¿R of the pinning interaction is constant, while the interface spacing T=TN is allowed to vary with the size N of the polymer. Our main result is the explicit determination of the scaling behavior of the model in the large N limit, as a function of (TN)N and for fixed d>0. In particular, we show that a transition occurs at TN=O(log¿N). Our approach is based on renewal theory.