  We study absolute continuity of harmonic measure with respect to surface
measure on domains $\Omega$ that have large complements. We show that if
$\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $
\mathbb{R}^{d+1}$ into two NTA domains then $\omega_{\Omega}\ll
\mathscr{H}^{d}$ on $\Gamma\cap \partial\Omega$. This result is a natural
generalisation of a result of Wu in [Wu86].
  We also prove that almost every point in $\Gamma\cap\partial\Omega$ is a cone
point if $\Gamma$ is a Lipschitz graph. Combining these results and a result
from [AHMMMTV], we characterize sets of absolute continuity with finite
$\mathscr{H}^{d}$-measure both in terms of the cone point condition and in
terms of the rectifiable structure of the boundary. This generalizes the
results of McMillan in [McM69] and Pommerenke in [Pom86].
  Finally, we also show our first result holds for elliptic measure associated
with real second order divergence form elliptic operators with a mild
assumption on the gradient of the matrix.
