  We compute the absorption spectrum of strongly repulsive one-dimensional
bosons in a disordered or quasi-periodic optical lattice. At commensurate
filling, the particle-hole resonances of the Mott insulator are broadened as
the disorder strength is increased. In the non-commensurate case, mapping the
problem to the Anderson model allows us to study the Bose-glass phase.
Surprisingly we find that a perturbative treatment in both cases, weak and
strong disorder, gives a good description at all frequencies. In particular we
find that the infrared absorption rate in the thermodynamic limit is quadratic
in frequency. This result is unexpected, since for other quantities like the
conductivity in one dimensional systems, perturbation theory is only applicable
at high frequencies. We discuss applications to recent experiments on optical
lattice systems, and in particular the effect of the harmonic trap.
