  We here conjecture that two much-studied aspects of quantum gravity,
dimensional flow and spacetime fuzziness, might be deeply connected. We
illustrate the mechanism, providing first evidence in support of our
conjecture, by working within the framework of multifractional theories, whose
key assumption is an anomalous scaling of the spacetime dimension in the
ultraviolet and a slow change of the dimension in the infrared. This sole
ingredient is enough to produce a scale-dependent deformation of the
integration measure with also a fuzzy spacetime structure. We also compare the
multifractional correction to lengths with the types of Planckian uncertainty
for distance and time measurements that was reported in studies combining
quantum mechanics and general relativity heuristically. This allows us to fix
two free parameters of the theory and leads, in one of the scenarios we
contemplate, to a value of the ultraviolet dimension which had already found
support in other quantum-gravity analyses. We also formalize a picture such
that fuzziness originates from a fundamental discrete scale invariance at short
scales and corresponds to a stochastic spacetime geometry.
