# Failure of rational approximation on some Cantor type sets

## TL;DR

This paper provides a counterexample demonstrating the failure of rational approximation equality on certain Cantor-type sets, and establishes conditions under which the equality holds, advancing understanding of function algebras on complex sets.

## Contribution

It constructs a counterexample to a question about algebra equality on specific compact sets and proves the equality under the condition that one set has no interior.

## Key findings

- Counterexample shows $R(K)
eq A(K)$ for certain Cantor-type sets.
- Equality $R(K) = A(K)$ holds if $K_1$ has no interior.
- Provides insight into rational approximation on complex fractal 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.