Universal spectra and Tijdeman's conjecture on factorization of cyclic groups

A spectral set Omega in R-n is a set of finite Lebesgue measure such that L -2(Omega) has an orthogonal basis of exponentials {e(2 pii(lambda ,x)) : la mbda is an element of Lambda} restricted to Omega. Any such set Lambda is c alled a spectrum for Omega. It is conjectured that every spectral set Omega tiles R-n by translations. A tiling set T of translations has a universal spectrum Lambda if every set Omega that tiles R-n by T is a spectral set wi th spectrum Lambda. Recently Lagarias and Wang showed that many periodic ti ling sets T have universal spectra. Their proofs used properties of factori zations of abelian groups, and were valid for all groups for which a strong form of a conjecture of Tijdeman is valid. However; Tijdeman's original co njecture is not true in general, as follows from a construction of Szabo [1 7], and here we give a counterexample to Tijdeman's conjecture for the cycl ic group of order 900. This article formulates a new sufficient condition f or a periodic tiling set to have a universal spectrum, and applies it to sh ow that the tiling sets in the given counterexample do possess universal sp ectra.