Shift invariant subspaces in growth spaces and sets of finite entropy

We investigate shift invariant subspaces within the realm of analytic functions in the unit disc, whose radial growth is determined by a majorant w$w$. Our main result offers a complete characterization of the shift invariant subspaces generated by Nevanlinna class functions within the aforementioned class of growth spaces. Our description is expressed in terms of a certain w$w$-entropy condition, which naturally arises in connection with boundary zero sets for analytic functions in the unit disc, having modulus of continuity not exceeding w$w$ on the unit circle. Furthermore, we utilize our findings to establish a structural theorem on approximation in model spaces.