  We consider a model for a polymer chain interacting with a sequence of
equispaced flat interfaces through a pinning potential. The intensity $\delta
\in \mathbb {R}$ of the pinning interaction is constant, while the interface
spacing $T=T_N$ 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 $(T_N)_N$ and for fixed $\delta >0$. In
particular, we show that a transition occurs at $T_N=O(\log N)$. Our approach
is based on renewal theory.
