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The measure matters

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.