Approximation by simple partial fractions with constraints on the poles

Under various constraints on a compact subset $K$ of the complex plane $\mathbb C$ and a subset $E\subset \mathbb C$ disjoint from $K$, the problem of density in the space $AC(K)$ (the space of functions that are continuous on a compact set $K$ and analytic in its interior) of the set of simple partial fractions (logarithmic derivatives of polynomials) with poles in $E$ is studied. The present investigation also involves examining some properties of additive subgroups of a Hilbert space.
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