  It is well-known that quantitative, scale invariant absolute continuity (more
precisely, the weak-$A_\infty$ property) of harmonic measure with respect to
surface measure, on the boundary of an open set $ \Omega\subset
\mathbb{R}^{n+1}$ with Ahlfors-David regular boundary, is equivalent to the
solvability of the Dirichlet problem in $\Omega$, with data in
$L^p(\partial\Omega)$ for some $p<\infty$. In this paper, we give a geometric
characterization of the weak-$A_\infty$ property, of harmonic measure, and
hence of solvability of the $L^p$ Dirichlet problem for some finite $p$. This
characterization is obtained under background hypotheses (an interior corkscrew
condition, along with Ahlfors-David regularity of the boundary) that are
natural, and in a certain sense optimal: we provide counter-examples in the
absence of either of them (or even one of the two, upper or lower,
Ahlfors-David bounds); moreover, the examples show that the upper and lower
Ahlfors-David bounds are each quantitatively sharp.
