The Corona Theorem on the Complements of Certain Square Cantor Sets

TL;DR

This paper proves the corona theorem for certain square Cantor set complements, showing it holds when the Cantor set's construction parameters decay sufficiently slowly.

Contribution

It establishes the corona theorem on domains formed by complements of specific square Cantor sets under new growth conditions.

Findings

Corona theorem holds for domains with Cantor sets where removal proportions decay as o(1/ log log n)

The result extends the class of domains where the corona theorem is valid

Provides new insights into the relationship between Cantor set construction and complex analysis

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