The corona theorem on the complement of certain square cantor sets

Let K be a square Cantor set, i.e., the Cartesian product K = E × E of two linear Cantor sets. Let δ _n denote the proportion of the intervals removed in the nth stage of the construction of E. It is shown that if $ \delta _n = o(\frac{1} {{\log \log n}}) $ \delta _n = o(\frac{1} {{\log \log n}}) , then the corona theorem holds on the domain Ω = ℂ^* \ K.