Motivated by an amazing result of D. Y. Grigoriev and M. Karpinski, the interpolation problem for k-sparse multivariate polynomials has received some attention in recent years. In this note we want to show that essentially all of the results obtained so far hold more generally for k-sparse sums of characters of abelian monoids, thereby providing a useful unified approach to this active field of research. As it turns out the basic ingredients of this approach are the construction of distinction sets for characters and zero-test sets for k-sparse character sums.