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

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

2000-08-16

arxiv.org/abs/math/0008132v1

J. Fourier Anal. Appl. 7 (2001), 63--70.

Mathematics

Functional Analysis

9 pages, latex, to appear J. Fourier Anal. Appl

Scientific paper

A spectral set in R^n is a set X of finite Lebesgue measure such that L^2(X) has an orthogonal basis of exponentials. It is conjectured that every spectral set tiles R^n 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.

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