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Under various constraints on a compact subset K of the complex plane C and a subset E ⊂ 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.Bibliography: 19 titles.