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Multi-fractional theories with integer-order derivatives are models ofgravitational and matter fields living in spacetimes with variable Hausdorffand spectral dimension, originally proposed as descriptions of geometriesarising in quantum gravity. We derive the Poisson equation and the Newtonianpotential of these theories starting from their covariant modified Einstein'sequations. In particular, in the case of the theory $T_v$ with weightedderivatives with small fractional corrections, we find a gravitationalpotential that grows logarithmically at large radii when the fractionalexponent takes the special value $\alpha=4/3$. This behaviour is associatedwith a restoration law for the Hausdorff dimension of spacetime independentlyfound in the dark-energy sector of the same theory. As an application, we checkwhether this potential can serve as an alternative to dark matter for thegalaxies NGC7814, NGC6503 and NGC3741 in the SPARC catalogue. We show thattheir rotation curves at medium-to-large radii can indeed be explained bypurely geometric effects, although the Tully--Fisher relation is not reproducedwell. We discuss how to fix the small-radius behaviour by lifting someapproximations and how to test the model with other observables and an enlargedgalaxy sample.

Gravitational potential and galaxy rotation curves in multi-fractional spacetimes