Spectra of certain types of polynomials and tiling of integers with   translates of finite sets
  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. 
Sergei Konyagin and Izabella Laba