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Let $\Omega\subset \mathbb{R}^{n+1}$ be an open set, not necessarilyconnected, with an $n$-dimensional uniformly rectifiable boundary. We show that$\partial\Omega$ may be approximated in a "Big Pieces" sense by boundaries ofchord-arc subdomains of $\Omega$, and hence that harmonic measure for $\Omega$is weak-$A_\infty$ with respect to surface measure on $\partial\Omega$,provided that $\Omega$ satisfies a certain weak version of a local Johncondition. Under the further assumption that $\Omega$ satisfies an interiorCorkscrew condition, and combined with our previous work, and with recent workof Azzam, Mourgoglou and Tolsa, this yields a geometric characterization ofdomains whose harmonic measure is quantitatively absolutely continuous withrespect to surface measure and hence a haracterization of the fact that theassociated $L^p$-Dirichlet problem is solvable for some finite $p$.

Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem. Part I