Cantor sets with absolutely continuous harmonic measure

We construct Ahlfors regular Cantor sets $K$ of small dimension in the plane, such that the Hausdorff measure on $K$ is equivalent to the harmonic measure associated to its complement. In particular the Green function in $R^2\mathrm{\backslash }K$ satisfies $G^p(x)\simeq \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,K{)}^{\delta }$ whenever $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,K)\le 1$ and $p$ is far from $K$.