  A spectral set in R^n is a set X of finite Lebesgue measure such that L^2(X)
has an orthogonal basis of exponentials. It is conjectured that every spectral
set tiles R^n by translations. A set of translations T has a universal spectrum
if every set that that tiles by translations by T has this spectrum. A recent
result proved that many periodic tiling sets have universal spectra, using
results from factorizations of abelian groups, for groups for which a strong
form of a conjecture of Tijdeman is valid. This paper shows Tijdeman's
conjecture does not hold for the cyclic group of order 900. It formulates a new
sufficient conjecture for a periodic tiling set to have a universal spectrum,
and uses it to show that the tiling sets for the counterexample above do have
universal spectra.
