Quantum-gravitational running/reduction of space-time dimension

Quantum-gravity renders the space-time dimension to depend on the size of region; it monotonically increases with the size of region and asymptotically approaches four for large distances. This effect was discovered in numerical simulations of lattice quantum gravity in the framework of causal dynamical triangulation [hep-th/0505113] as well as in renormalization group approach to quantum gravity [hep-th/0508202]. However, along these approaches the interpretation and the physical meaning of the effective change of dimension at shorter scales is not clear. Without invoking particular models in this essay we show that, box-counting dimension in face of finite resolution of space-time (generally implied by quantum gravity) shows a simple way how both the qualitative and the quantitative features of this effect can be understood. In this way we derive a simple analytic expression of space-time dimension running, which implies the modification of Newton's inverse square law in a perfect agreement with the modification coming from one-loop gravitational radiative corrections.