  We adopt a framework where quantum-gravity's dynamical dimensional reduction
of spacetime at short distances is described in terms of modified dispersion
relations. We observe that by subjecting such models to a momentum-space
diffeomorphism one obtains a "dual picture" with unmodified dispersion
relations, but a modified measure of integration over momenta. We then find
that the UV {\it Hausdorff} dimension of momentum space which can be inferred
from this modified integration measure coincides with the short-distance {\it
spectral} dimension of spacetime. This result sheds light into why
scale-invariant fluctuations are obtained if the original model for two UV
spectral dimensions is combined with Einstein gravity. By studying the
properties of the inner product we derive the result that it is only in 2
energy-momentum dimensions that microphysical vacuum fluctuations are
scale-invariant. This is true ignoring gravity, but then we find that if
Einstein gravity is postulated in the original frame, in the dual picture
gravity switches off, since all matter becomes conformally coupled. We argue
that our findings imply that the following concepts are closely connected:
scale-invariance of vacuum quantum fluctuations, conformal invariance of the
gravitational coupling, UV reduction to spectral dimension 2 in position space
and UV reduction to Hausdorff dimension 2 in energy-momentum space.
