Universal Spectra and Tijdeman's Conjecture on Factorization of Cyclic Groups

Abstract

A spectral set in R n is a set of finite Lebesgue measure such that L 2 () has an orthogonal basis of exponentials {e 2�ih �,xi : � ∈ �} restricted to . Any such setis called a spectrum for . It is conjectured that every spectral set tiles R n by translations. A tiling set T of translations has a universal spectrumif every set that tiles R n by T is a spectral set with spectrum �. Recently Lagarias and Wang showed that many periodic tiling sets T have universal spectra. Their proofs used properties of factorizations of abelian groups, and were valid for all groups for which a strong form of a conjecture of Tijdeman is valid. However Tijdeman's original conjecture is not true in general, as follows from a construction of Szabo (17), and here we give a counterexample to Tijdeman's conjecture for the cyclic group of order 900. This paper formulates a new sufficient conditionfor a periodic tiling set to have a universal spectrum, and applies it to show that the tiling sets in the given counterexample do possess universal spectra.