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/N<inf>1</inf>)Z×...×(1/N<inf>n</inf>)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 spectrumΛ<inf>S</inf>. We callΛ<inf>S</inf>a 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=Z<inf>N1</inf>×...×Z<inf> Nn</inf>, andA:=A (modZⁿ). In dimensionn=1 we show that S has a universal spectrum wheneverN<inf>1</inf>is the order of a "good" group in the sense of Hajós, and for various other sets S. © 1997 Academic Press.