Failure of rational approximation on some Cantor type sets

Abstract:

Let $A(K)$ be the algebra of continuous functions on a compact set $K\subset \mathbb {C}$ which are analytic on the interior of $K$, and let $R(K)$ be the closure (with respect to uniform convergence on $K$) of the functions that are analytic on a neighborhood of $K$. A counterexample of a question posed by A. O’Farrell about the equality of the algebras $R(K)$ and $A(K)$ when $K=(K_{1}\times [0,1])\cup ([0,1]\times K_{2})\subseteq \mathbb {C}$, with $K_{1}$ and $K_{2}$ compact subsets of $[0,1]$, is given. Also, the equality is proved with the assumption that $K_{1}$ has no interior.