is here that we need the condition that $\delta_n=o(\frac1{\log\log n})$. Section 3 is devoted to constructing these interpolating functions and solving the associated $\bar\partial$-problem in our simply connected subdomains.

Once these solutions have been constructed we paste these solutions together by solving another $\bar\partial$-problem. In this case, the $\bar\partial$-problem is solved using the ideas of rational approximation theory (see [@vitushkin] or Chapter XII of[@gamelin-ua]). Essentially, one solves the $\bar\partial$-problem on the intersection of two subdomains by the usual Cauchy integral representation, but in order to be able to sum these various pieces, we must add additional cancellation to each piece. This is done by subtracting a bounded analytic function on $\Omega$ which has singular support on a portion of the Cantor set nearby and which matches derivatives of the integral at infinity. Schwarz lemma bounds, in conjunction with the cancellations from our special corona solutions in the subdomains, then allow us to sum the terms and obtain the solution to the $\bar\partial$-problem. This part of the proof implicitly makes use of the fact that for any $\zeta\in K$ there is $c_0>0$ such that $$\gamma(B(\zeta,r)\cap K)\ge c_0r$$ for $r\in(0,{{\rm{diam}}}(K)]$, where $$\gamma(E):=\sup\{|f'(\infty)|:f\in H^\infty(\mathbb C^\ast\setminus E),\Vert f\Vert\le1\}$$ denotes the analytic capacity of the 