  We show that the uncertainty in distance and time measurements found by the
heuristic combination of quantum mechanics and general relativity is reproduced
in a purely classical and flat multi-fractal spacetime whose geometry changes
with the probed scale (dimensional flow) and has non-zero imaginary dimension,
corresponding to a discrete scale invariance at short distances. Thus,
dimensional flow can manifest itself as an intrinsic measurement uncertainty
and, conversely, measurement-uncertainty estimates are generally valid because
they rely on this universal property of quantum geometries. These general
results affect multi-fractional theories, a recent proposal related to quantum
gravity, in two ways: they can fix two parameters previously left free (in
particular, the value of the spacetime dimension at short scales) and point
towards a reinterpretation of the ultraviolet structure of geometry as a
stochastic foam or fuzziness. This is also confirmed by a correspondence we
establish between Nottale scale relativity and the stochastic geometry of
multi-fractional models.
