Galaxy rotation curves in multi-fractional spacetimes
  Multi-fractional theories with integer-order derivatives are simple models of
gravitational and matter fields living in spacetimes with variable Hausdorff
and spectral dimension, originally proposed as descriptions of spacetimes
arising in quantum gravity. In this paper, we pose the question of whether they
can serve as an alternative to dark matter. We give a preliminary positive
answer. We find the Poisson equation and the Newtonian potential of the
multi-fractional theories with integer-order derivatives starting from their
covariant modified Einstein's equations. We show that neither the theory $T_v$
with weighted derivatives in the extreme fractional limit nor the theory $T_q$
with $q$-derivatives fit the rotation curve of any of the galaxies NGC7814,
NGC6503 and NGC3741 in the SPARC catalogue. The rotation curve of all three
galaxies can be explained by purely geometric effects in the theory $T_v$ with
weighted derivatives with small fractional corrections when the fractional
exponent takes the special value $\alpha=4/3$, but only at large radii.
