  Multi-fractional theories with integer-order derivatives are models of
gravitational and matter fields living in spacetimes with variable Hausdorff
and spectral dimension, originally proposed as descriptions of geometries
arising in quantum gravity. We derive the Poisson equation and the Newtonian
potential of these theories starting from their covariant modified Einstein's
equations. In particular, in the case of the theory $T_v$ with weighted
derivatives with small fractional corrections, we find a gravitational
potential that grows logarithmically at large radii when the fractional
exponent takes the special value $\alpha=4/3$. This behaviour is associated
with a restoration law for the Hausdorff dimension of spacetime independently
found in the dark-energy sector of the same theory. As an application, we check
whether this potential can serve as an alternative to dark matter for the
galaxies NGC7814, NGC6503 and NGC3741 in the SPARC catalogue. We show that
their rotation curves at medium-to-large radii can indeed be explained by
purely geometric effects, although the Tully--Fisher relation is not reproduced
well. We discuss how to fix the small-radius behaviour by lifting some
approximations and how to test the model with other observables and an enlarged
galaxy sample.
