B-Points of a Cantor-Type Set

We study the behavior of the harmonic continuation u to the upper half-plane of the characteristic function for a Cantor-type set E of positive length, which is precisely the harmonic measure of such a set, as it approaches the boundary. We are interested in describing the points x ∈ E, given in terms of their Cantor encoding, such that the mean variation of u along the segment [x, x + i], i.e., a certain weighted average of the variations along [x, x + t + i], t ∈ ℝ, is finite.