We compare the recently formulated multifractional spacetimes with field
theories of quantum gravity based on the renormalization group (RG), such as
asymptotic safety and Ho\v{r}ava--Lifshitz gravity. The change of spacetime
dimensionality with the probed scale is realized in both cases by an adaptation
of the measurement tools ("rods") to the scale, but in different ways. In the
multifractional case, by an adaptation of the position-space measure, which can
be encoded into an explicit scale dependence of effective coordinates. In the
case of RG-based theories, by an adaptation of the momenta. The two pictures
are mapped into each other, thus presenting the fractal structure of spacetime
in RG-based theories under an alternative perspective.Comment: 21 pages, 1 table. v2: discussion improved, references adde
