  Analogs of classical results on accumulated spectrograms, the sum of
spectrograms of eigenfunctions of localization operators, are established for
Gabor multipliers on tight frames. We show that the lattice $\ell^1$ distance
between the accumulated spectrogram and the indicator function of the Gabor
multiplier mask is bounded by the length of the perimeter of the mask and that
this bound is sharp in general. The methods developed for the proofs are also
used to show that the Weyl-Heisenberg ensemble restricted to a lattice is
hyperuniform.
