2000/08/16 by Jeffrey C. Lagarias, Sandor Szabo
Mathematics · #math.FA #math.SP #msc:47A13 #msc:11K70 #msc:42B05
published as J. Fourier Anal. Appl. 7 (2001), 63--70. · 9 pages, latex, to appear J. Fourier Anal. Appl
arxiv created 2000/08/16 · arxiv updated 2009/11/30
A spectral set in Rn is a set X of finite Lebesgue measure such that L2(X) has an orthogonal basis of exponentials. It is conjectured that every spectral set tiles Rn by translations. A set of translations T has a universal spectrum if every set that that tiles by translations by T has this spectrum. A recent result proved that many periodic tiling sets have universal spectra, using results from factorizations of abelian groups, for groups for which a strong form of a conjecture of Tijdeman is valid. This paper shows Tijdeman's conjecture does not hold for the cyclic group of order 900. It formulates a new sufficient conjecture for a periodic tiling set to have a universal spectrum, and uses it to show that the tiling sets for the counterexample above do have universal spectra.