  We examine vacuum fluctuations in theories with modified dispersion relations
which represent dimensional reduction at high energies. By changing units of
energy and momentum we can obtain a description rendering the dispersion
relations undeformed and transferring all the non-trivial effects to the
integration measure in momentum space. Using this description we propose a
general quantization procedure, which should be applicable whether or not the
theory explicitly introduces a preferred frame. Based on this scheme we
evaluate the power spectrum of quantum vacuum fluctuations. We find that in
{\it all} theories which run to 2 dimensions in the ultraviolet the vacuum
fluctuations, in the ultraviolet regime, are scale-invariant. This is true in
flat space but also for "inside the horizon" modes in an expanding universe. We
spell out the conditions upon the gravity theory for this scale-invariance to
be preserved as the modes are frozen-in outside the horizon. We also digress on
the meaning of dimensionality (in momentum and position space) and suggest that
the spectral index could itself provide an operational definition of
dimensionality.
