Approximations by simple partial fractions with constraints on the poles

We discusss the following new result: if the plane compact set $K$ has connected complement and is covered by the union $\widehat{E}\setminus E$ of bounded components of complement of another compact set $E$, then the simple partial fractions (logarithmic derivatives of polynomials) having all poles on $E$ are dense in the space $AC(K)$ of functions that are continuous on $K$ and analytic on its interior.