  We calculate the interaction induced inhomogeneous broadening of spectral
lines in a trapped Bose gas as a function of the depth of a three-dimensional
cubic optical lattice. As observed in recent experiments, we find that the
terraced "wedding-cake" structure of Mott plateaus splits the spectrum into a
series of discrete peaks. The spectra are extremely sensitive to density
corrugations and trap anharmonicities. For example, even when the majority of
the cloud is superfluid the spectrum displays discrete peaks.
