  We consider an evaporating Schwarzschild black hole in a framework in which
the spectral dimension of spacetime varies continuously from four at large
distances to a number smaller than three at small distances, as suggested by
various approaches to quantum gravity. We demonstrate that the evaporation
stops when the horizon radius reaches a scale at which spacetime becomes
effectively 3-dimensional, and argue that an observer remaining outside the
horizon cannot probe the properties of the black hole at smaller scales. This
result is universal in the sense that it does not depend on the details of the
effective dimension as a function of the diffusion time. Observers falling into
the black hole can resolve smaller scales, as can external observers in the
presence of a cosmological constant. Even in these cases, though, we obtain an
absolute bound D>2 on the effective dimension that can be seen in any such
attempt to measure the properties of the black hole.
