  We revisit one of the earliest proposals for deformed dispersion relations in
the light of recent results on dynamical dimensional reduction and production
of cosmological fluctuations. Depending on the specification of the measure of
integration and addition rule in momentum space the model may be completed so
as to merely deform Lorentz invariance, or so as to introduce a preferred
frame. Models which violate Lorentz invariance have a negative UV asymptotic
dimension and a very red spectrum of quantum vacuum fluctuations. Instead,
models which preserve frame independence can exhibit running to a UV dimension
of 2, and a scale-invariant spectrum of fluctuations. The bispectrum of the
fluctuations is another point of divergence between the two casings proposed
here for the original model.
