Gravitational potential and galaxy rotation curves in multi-fractional spacetimes

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