On the universal tiling conjecture in dimension one
  We show that the spectral-tile implication in the Fuglede conjecture in dimension 1 is equivalent to a Universal Tiling Conjecture and also to similar forms of the same implication for some simpler sets, such as unions of intervals with rational or integer endpoints. 
Dorin Ervin Dutkay and Palle E. T. Jorgensen
math.FA