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

Square functions, non-tangential limits and harmonic measure in co-dimensions larger than one