We show that for uniform domains whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to -dimensional Hausdorff measure unless . 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.
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Jonas Azzam, Mihalis Mourgoglou, Tangent measures and absolute continuity of harmonic measure. Rev. Mat. Iberoam. 34 (2018), no. 1, pp. 305–330DOI 10.4171/RMI/986