Spectral Sets and Factorizations of Finite Abelian Groups

A spectral set is a subset 0 of R with Lebesgue measure 0<+(0)< such that there exists a set 4 of exponential functions which form an orthogonal basis of L(0). The spectral set conjecture of B. Fuglede states that a set 0 is a spectral set if and only if 0 tiles R by translation. We study sets 0 which tile R using a rational periodic tile set S=Z+A, where A (1 N1)Z_ } } } _(1 Nn)Z is finite. We characterize geometrically bounded measurable sets 0 that tile R with such a tile set. Certain tile sets S have the property that every bounded measurable set 0 which tiles R with S is a spectral set, with a fixed spectrum 4S . We call 4S a universal spectrum for such S. We give a necessary and sufficient condition for a rational periodic set 4 to be a universal spectrum for S, which is expressed in terms of factorizations A B=G where G=ZN1_ } } } _ZNn , and A := A (mod Z). In dimension n=1 we show that S has a universal spectrum whenever N1 is the order of a ``good'' group in the sense of Hajo s, and for various other sets S. 1997 Academic Press