Spectral Sets and Factorizations of Finite Abelian Groups

Abstract

Aspectral setis a subsetΩofRnwith Lebesgue measure 0<μ(Ω)<∞ such that there exists a setΛof exponential functions which form an orthogonal basis ofL2(Ω). The spectral set conjecture of B. Fuglede states that a set 0 is a spectral set if and only ifΩtilesRnby translation. We study setsΩwhich tileRnusing a rational periodic tile set S=Zn+A, where A⊆(1/N1)Z×…×(1/Nn)Zis finite. We characterize geometrically bounded measurable setsΩthat tileRnwith such a tile set. Certain tile sets S have the property that every bounded measurable setΩwhich tilesRnwith S is a spectral set, with a fixed spectrumΛS. We callΛSa universal spectrum for such S. We give a necessary and sufficient condition for a rational periodic setΛto be a universal spectrum for S, which is expressed in terms of factorizationsA⊕B=GwhereG=ZN1×…×ZNn, andA:=A (modZn). In dimensionn=1 we show that S has a universal spectrum wheneverN1is the order of a “good” group in the sense of Hajós, and for various other sets S.