Spectra of certain types of polynomials and tiling of integers with translates of finite sets

We investigate Fugledeʹs spectral set conjecture in the special case when the set in question is a union of finitely many unit intervals in dimension 1. In this case, the conjecture can be reformulated as a statement about multiplicative properties of roots of associated with the set polynomials with (0,1) coefficients. Let A(x) z[x] be an N-term polynomial. We say that {θ1,θ2,…,θN−1} is an N-spectrum for A(x) if the θj are all distinct and We establish necessary and sufficient conditions for irreducible polynomials and for products of two factors of the form 1+xk+x2k+…+xmk to have a spectrum. This confirms Fugledeʹs conjecture for associated sets.