  We consider two number-theoretic problems arising from Fuglede's spectral set
conjecture: characterizing finite sets that tile integers, and finding
polynomials with (0,1) coefficients whose roots have a certain multiplicative
structure. We verify several special cases of the relevant conjectures; in
particular, we find necessary and sufficient conditions for a set A to tile the
integers if A is a direct sum of cyclic subsets.
