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 $R(K)$ the closure (with the uniform convergence on $K$) of the functions that are analytic on a neighborhood of $K$. A counterexample of a question made 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.

6 pages, 1 figure