  We study the topological gap labeling of general 3D quasicrystals and we find
that every gap in the spectrum is characterized by a set of the third Chern
numbers. We show that a quasi-periodic structure has multiple Brillouin zones
defined by redundant wavevectors, and the number of states below a gap is
quantized as an integer linear combination of volumes of these Brillouin zones.
The associated quantum numbers to characterize energy gaps can be expressed as
third Chern numbers by considering a formal relationship between an adiabatic
charge pumping under cyclic deformation of the quasi-periodic potential and a
topological nonlinear electromagnetic response in 6D insulators.
