  In this paper, we characterize the rectifiability (both uniform and not) of
an Ahlfors regular set, E, of arbitrary co-dimension by the behavior of a
regularized distance function in the complement of that set. In particular, we
establish a certain version of the Riesz transform characterization of
rectifiability for lower-dimensional sets. We also uncover a special situation
in which the regularized distance is itself a solution to a degenerate elliptic
operator in the complement of E. This allows us to precisely compute the
harmonic measure of those sets associated to this degenerate operator and prove
that, in a sharp contrast with the usual setting of co-dimension one, a
converse to the Dahlberg's theorem (see [Da] and [DFM2]) must be false on lower
dimensional boundaries without additional assumptions.
