  We study zero sets of twisted stationary Gaussian random functions on the
complex plane, i.e., Gaussian random functions that are stochastically
invariant under the action of the Weyl-Heisenberg group. This model includes
translation invariant Gaussian entire functions (GEFs), and also many other
non-analytic examples, in which case winding numbers around zeros can be either
positive or negative. We investigate zero statistics both when zeros are
weighted with their winding numbers (charged zero set) and when they are not
(uncharged zero set).
  We show that the variance of the charged zero statistic always grows linearly
with the radius of the observation disk (hyperuniformity). Importantly, this
holds for functions with possibly non-zero means and without assuming
additional symmetries such as radiality. With respect to uncharged zero
statistics, we provide an example for which the variance grows with the area of
the observation disk (non-hyperuniformity). This is used to show that, while
the zeros of GEFs are hyperuniform, the set of their critical points fails to
be so.
  Our work contributes to recent developments in statistical signal processing,
where the time-frequency profile of a non-stationary signal embedded into noise
is revealed by performing a statistical test on the zeros of its spectrogram
(``silent points''). We show that empirical spectrogram zero counts enjoy
moderate deviation from their ensemble averages over large observation windows
(something that was previously known only for pure noise). In contrast, we also
show that spectogram maxima (``loud points") fail to enjoy a similar property.
This gives the first formal evidence for the statistical superiority of silent
points over the competing feature of loud points, a fact that has been noted by
practitioners.
