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abstract: 'Let $K$ be a square Cantor set, i.e. the Cartesian product $K=E\times E$ of two linear Cantor sets. Let $\delta_n$ denote the proportion of the intervals removed in the $n$th stage of the construction of $E$. It is shown that if $\delta_n=o(\frac1{\log\log n})$ then the corona theorem holds on the domain $\Omega=\mathbb C^\ast\setminus K$.'
author:
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    Jon Handy\
    University of California, Los Angeles
title: The Corona Theorem on the Complement of Certain Square Cantor Sets
---

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Introduction
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Denote by $H^\infty(\Omega)$ the collection of bounded analytic functions on a plane domain $\Omega$. Let us denote by $\mathcal M=\mathcal M(H^\infty(\Omega))$ the set of multiplicative linear functionals on $H^\infty(\Omega)$, i.e. $\mathcal M$ is the maximal ideal space of $H^\infty(\Omega)$. When $H^\infty(\Omega)$ separates the points of $\Omega$, there is a natural identification of $\Omega$ with a subset of $\Omega$ through the functionals defined by pointwise evaluations. The corona problem for $H^\infty(\Omega)$ is to determine whether $\mathcal M$ is the closure of $\Omega$ in the Gelfand topology.

The problem can be stated more concretely as follows. Given $\{f_j\}_1^n\in H^\infty(\Omega)$ and $\delta>0$ such that $1\ge\max_j|f_j(z)|>\delta$ for all $z\in\Omega$, do there exist $\{g_j\}_1^n\in H^\infty(\Omega)$ such 