Tangent measures and absolute continuity of harmonic measure

We show that for uniform domains Omega subset of Rd+1 whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to alpha-dimensional Hausdorff measure unless alpha <= d. We employ a lemma that shows that, at almost every non-degenerate point, we may find a tangent measure of harmonic measure whose support is the boundary of yet another uniform domain whose harmonic measure resembles the tangent measure.