  We establish the equivalence between the regularity (rectifiability) of sets
and suitable estimates on the oscillation of the gradient for smooth non-local
distance functions. A prototypical example of such a distance was introduced,
as part of a larger PDE theory, by Guy David, Joseph Feneuil, and the third
author. The results apply to all dimensions and co-dimensions, require no
underlying topological assumptions, and provide a surprisingly rich class of
analytic characterizations of rectifiability.
