  Several approaches to quantum gravity suggest that the standard description
of spacetime as probed at low-energy, with four dimensions, is replaced in the
Planckian regime by a spacetime with a spectral dimension of two. The
implications for relativistic symmetries can be momentous, and indeed the most
tangible picture for "running" of the spectral dimension, found within
Horava-Lifschitz gravity, requires the breakdown of relativity of inertial
frames. In this Letter we incorporate running spectral dimensions in a scenario
that does not require the emergence of a preferred frame. We consider the best
studied mechanism for deforming relativistic symmetries whilst preserving the
relativity of inertial frames, based on a momentum space with curvature at the
Planck scale. We show explicitly how running of the spectral dimension can be
derived from these models.
