Low dimensional Cantor sets with absolutely continuous harmonic measure

The relationship between harmonic measure and surface measure of a domain is largely connected with the geometry of the domain itself. In many fractals (for example, in domains with relatively "large'' boundaries, and outside self-similar "enough'' Cantor sets), these measures are mutually singular, and in fact, have different dimensions. After recalling some of these results I will present joint work with G. David and A. Julia where we demonstrate examples where the exact opposite occurs: we construct Cantor-type sets in the plane that are Ahlfors regular (of small dimension) for which their associated harmonic measure and surface measure are bounded equivalent.