  We show that the dimension of spacetime becomes complex-valued when its
short-scale geometry is invariant under a discrete scaling symmetry. This
characteristic can generically arise in quantum gravities, for instance, in
those based on combinatorial or multifractal structures or as the partial
breaking of continuous dilation symmetry in any conformal-invariant theory.
With its infinite scale hierarchy, discrete scale invariance overlaps with the
traditional separation between ultraviolet and infrared physics and it can
leave an all-range observable imprint, such as a pattern of log oscillations
and sharp features in the cosmic microwave background primordial power
spectrum.
