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 ${\delta }_n=o(\frac{1}{\log \log n})$ then the corona theorem holds on the domain $\Omega ={\mathbb{C}}^{\ast }\setminus K$.

16 pages, 3 figures. (submitted: Journal d'Analyse Mathematique)