set $E$ (see [@tmv]). This parallels the thickness condition for the boundary employed in [@car-fatsets]. Thus the functions introduced to obtain cancellations are polynomials in the extremal function for this problem, the Ahlfor’s function, for a piece of the boundary. Alternatively, one can take the function to be powers of the Cauchy integral of the uniform measure on an appropriate subsquare $K^n_J:=K\cap Q^n_J$ of the Cantor set (see [@garnett-ac]).

The present work is part of the author’s Ph.D. dissertation. The author is naturally greatly indebted to his advisor, John Garnett, for many helpful conversations and suggestions over the years, and would like to take this opportunity to express his gratitude. The author would also like to thank Peter Jones for helpful conversations.

Proof of the Theorem
====================

To begin the proof, let us first define simply connected, cross-shaped regions $T^n_J$ by $T^n_J:=Q^n_J\setminus\bigcup_1^4Q^{n+1}_{J,j}$. For these regions we distinguish the four boundary segments, denoted $\{\ell^n_{J,j}\}_{j=1}^4$, which do not lie on a side of a square $Q^{n+1}_{J,j}$, $1\le j\le4$. Let us then define lozenges $$\label{lozenges}
    L^n_{J,j}=\left\{z:\left|\arg\frac{z-x_2}{x_2-x_1}\right|\vee\left|\arg\frac{z-x_1}{x_2-x_1}\right|<\alpha\right\},$$ where $(x_1,x_2)=\ell^n_{J,j}$ and $\alpha$ is some angle less than $\frac\pi4$ to be determined by the constructions below. We now define regions $$\hat T^n_J=T^n_J\cup\bigcup\{L^m_{I,i}:\ell^m_{I,i}\subset\partial T^n_J\}.$$ (See Figure \[fig2\].) 