article no. FU963008 Spectral Sets and Factorizations of Finite Abelian Groups

A spectral set is a subset 0 of R n with Lebesgue measure 0<+(0)< such that there exists a set 4 of exponential functions which form an orthogonal basis of L 2 (0). The spectral set conjecture of B. Fuglede states that a set 0 is a spectral set if and only if 0 tiles R n by translation. We study sets 0 which tile R n using a rational periodic tile set S=Z n +A, where A (1 N 1)Z_}}}_(1 N n)Z is finite. We characterize geometrically bounded measurable sets 0 that tile R n with such a tile set. Certain tile sets S have the property that every bounded measurable set 0 which tiles R n with S is a spectral set, with a fixed spectrum 4 S. We call 4 S 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=Z N1 _}}}_Z Nn, and A:= A (mod Z n). In dimension n=1 we show that S has a universal spectrum whenever N 1 is the order of a ``good' ' group in the sense of Hajos, and for various other sets S. 1997 Academic Press 1