SPECTRAL SETS AND FACTORIZATIONS OF FINITE ABELIAN-GROUPS

A spectral set is a subset Ohm of R(n) with Lebesgue measure 0 <mu(Ome ga)< infinity such that there exists a set A of exponential functions which form an orthogonal basis of L(2)(Omega). The spectral set conjec ture of B. Fuglede states that a set Omega is a spectral set if and on ly if Omega tiles R(n) by translation. We study sets Omega which tile R(n) using a rational periodic tile set S = Z(n) + A, where A subset o f or equal to(1/N-1)Zx ... x(1/N-n)Z is finite. We characterize geomet rically bounded measurable sets Omega that tile R(n) with such a tile set. Certain tile sets S have the property that every bounded measurab le set Omega which tiles R(n) with S is a spectral set, with a Fixed s pectrum A(S). We call A(S) a universal spectrum for such S. We give a necessary and sufficient condition for a rational periodic set A to be a universal spectrum for S, which is expressed in terms of Factorizat ions A+B=G where G=Z(N1)x ... xZN(n), and A:= A (mod Z(n)). In dimensi on n=1 we show that S has a universal spectrum whenever N-1 is the ord er of a ''good'' group in the sense of Hajos, and for various Other se ts S. (C) 1997 Academic Press.