BMO rational approximation and one-dimensional Hausdorff content

Let $X \subset {\mathbf{C}}$ be compact and let $f \in \operatorname{VMO} ({\mathbf{C}})$. We give necessary and sufficient conditions on $f$ and $X$ for ${f_{\vert X}}$ to be the limit of a sequence of rational functions without poles on $X$ in the norm of $\operatorname{BMO} (X)$, the space of functions of bounded mean oscillation on $X$. We also characterize those compact $X \subset {\mathbf{C}}$ with the property that the restriction to $X$ of each function in $\operatorname{VMO} ({\mathbf{C}})$, which is holomorphic on $\mathop X\limits^ \circ$, is the limit of a sequence of rational functions with poles off $X$. Our conditions involve the notion of one-dimensional Hausdorff content. As an application, a result related to the inner boundary conjecture is proven.