Mean Rational Approximation for Compact Subsets with Thin Boundaries

In 1991, J. Thomson obtained a celebrated decomposition theorem for $P^t(\mu ),$ the closed subspace of $L^t(\mu )$ spanned by the analytic polynomials, when $1\le t<\mathrm{ı}\mathrm{.}$ In 2008, J. Brennan \cite{b08} generalized Thomson’s theorem to $R^t(K,\mu ),$ the closed subspace of $L^t(\mu )$ spanned by the rational functions with poles off a compact subset $K$ containing the support of $\mu ,$ when the diameters of the components of $\mathbb{C}\setminus K$ are bounded below. We extend the above decomposition theorems for $R^t(K,\mu )$ when the boundary of $K$ is not too wild.