We study the structure of the digit sets {\mathcal D} for the integral self-similar tiles T(b,{\mathcal {D}}) (we call such a {\mathcal D} a tile digit set with respect to b). So far the only available classes of such tile digit sets are the complete residue sets and the product-forms. Our investigation here is based on the spectrum of the mask polynomial P_{\mathcal D}, i.e., the zeros of P_{\mathcal D} on the unit circle. By using the Fourier criteria of self-similar tiles of Kenyon and Protasov, as well as the algebraic techniques of cyclotomic polynomials, we characterize the tile digit se...
