Mean Rational Approximation for Some Compact Planar Subsets

In 1991, J. Thomson obtained celebrated structural results for $P^t(\mu).$ Later, J. Brennan (2008) generalized Thomson's theorem to $R^t(K,\mu)$ when the diameters of the components of $\mathbb C\setminus K$ are bounded below. The results indicate that if $R^t(K,\mu)$ is pure, then $R^t(K,\mu) \cap L^\infty (\mu)$ is the "same as" the algebra of bounded analytic functions on $\mbox{abpe}(R^t(K, \mu)),$ the set of analytic bounded point evaluations. We show that if the diameters of the components of $\mathbb C\setminus K$ are allowed to tend to zero, then even though $\text{int}(K) = \mbox{abpe}(R^t(K, \mu))$ and $K =\overline {\text{int}(K)},$ the algebra $R^t(K,\mu) \cap L^\infty (\mu)$ may "be equal to" a proper sub-algebra of bounded analytic functions on $\text{int}(K),$ where functions in the sub-algebra are "continuous" on certain portions of the inner boundary of $K.$