$B$-points of a Cantor-type set

In this note we study the behavior of the harmonic continuation $u$ to the upper half-plane for the characteristic function of a Cantor-type set $E$ of positive length, which is precisely the harmonic measure of such a set, near the boundary. We are interested in the description of points $x\in E$ (given in terms of their Cantor encoding) such that the mean variation of $u$ along $[x,x+i]$ – a certain weighted average of variations along $[x,x+t+i]$, $t\in\mathbb{R}$ – is finite.