Cantor sets with absolutely continuous harmonic measure
Guy David, Cole Jeznach, Antoine Julia · Journal de l’École polytechnique — Mathématiques · 2023
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 Green’s function in ℝ 2 ∖ K satisfies G p ( x ) ≃ dist ( x , K ) δ whenever dist ( x , K ) ≤ 1 and p is far from K .