is a multi-index of length $|J|=n$ on letters $\{1,2,3,4\}$. At stage $n+1$, we construct squares $Q^{n+1}_{J,j}\subset Q^n_J$, $j\in\{1,2,3,4\}$, of side length $\sigma_{n+1}$ with sides parallel to the axes such that each $Q^{n+1}_{J,j}$, $1\le j\le4$, contains a corner of $Q^n_J$. We define $K^{n+1}=\bigcup_{|J|=n}\bigcup_{j=1}^4Q^{n+1}_{J,j}$.

Some auxiliary definitions are also useful in this setting. The first of these is the quantity $\delta_n=1-2\lambda_n$, which represents the normalized gap width between squares of the $n$th generation having common parent. It is useful also to introduce the thickened squares $V^n_J:=(1+\frac{\delta_n}{2\lambda_n})Q^n_J$, and from these the “square annuli” $A^n_J$ given by $A^n_J=\overline{V^n_J\setminus\bigcup_{j=1}^4V^{n+1}_{J,j}}$. We note that our constants were chosen so that $\bigcup_1^4V^{n+1}_{J,j}$ is a single square concentric with and containing $Q^n_J$. That $A^n_J\neq\emptyset$ follows from the assumption that $\lambda_n\le\lambda_{n+1}$. Let us denote by $z^n_J$ the center of the square $Q^n_J$. (See Figure \[fig1\].)

For the harmonic measures below, we will generally make the abbreviations $\omega(\cdot):=\omega(\infty,\cdot,\Omega)$ and $\omega(\cdot,z):=\omega(z,\cdot,\Omega)$ or $\omega_z(\cdot):=\omega(z,\cdot,\Omega)$ when there is no risk of confusion.

We briefly sketch the proof of the result of Jones. Let us denote by $\{z_j\}$ the critical points of Green’s function $g(z)=g(z,\infty)$ for the domain $\Omega$. For $\{z_j\}$ to be an interpolating sequence, it is necessary that the sum $\sum g(z_j)$ be 