We know that a set may be of zero harmonic measure without its Lebesgue measure being zero (see Armitage and Gardiner, classical potential theory, pg 178).

Now, consider the following problem. Let $ V $ be a bounded open set in $ \mathbb{R}^{m} $, $ m\geq 2 $, and $W$ the interior of the closure of $V$. Let $E$ be a subset of $ \partial V\cap W $ ($\partial$ means boundary). We suppose $E$ has the following property: for every $x\in E$ and every $r>0$ such that the ball $B(x,r)$ is relatively compact in $W$, the boundary $\partial B(x,r)$ of $B(x,r)$ contains at least one point of $E$. My question is: can the harmonic measure $\omega(E,x,V)$ of $E$ be zero?

Intuitively (Brownian motion interpretation of harmonic measure) the answer seems negative. (?)