  Quasiperiodic systems show a universal gap structure due to quasiperiodicity
which is analogous to gap openings at the Brillouin zone boundary in periodic
systems. The integrated density of states (IDoS) below those energy gaps are
characterized by a few integers, which is known as the ``gap labeling theorem''
(GLT) for quasiperiodic systems. In this study, focusing on multilayer thin
film systems such as twisted bilayer graphene and stacked transition metal
dichalcogenides, we extend the GLT for multilayer systems of arbitrary
dimensions and number of layers, using an approach based on the algebra called
``a noncommutative torus''. We find that the energy gaps and the associated
IDoS are generally characterized by $_{DN}C_D$ integer labels in $N$ layer
systems in the $D$ dimensions, when the effect of the interlayer coupling can
be approximated by a quasiperiodic intralayer coupling for each layer. We
demonstrate that the generalized GLT holds for quasiperiodic 1D tight binding
models by numerical simulations.
