Sets of Hilbert series and their applications

We consider graded finitely presented algebras and modules over a field. Under some restrictions, the set of Hilbert series of such algebras (or modules) becomes finite. Claims of that type imply the rationality of Hilbert and Poincaré series of some algebras and modules, including the periodicity of Hilbert functions of many (e.g., Noetherian) modules and algebras of linear growth.