The Corona Theorem on the Complements of Certain Square Cantor Sets

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 $\Omega=\mathbb C^\ast\setminus K$.