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Universal Spectra and Tijdeman's Conjecture on Factorization of Cyclic Groups | | Jeffrey C. Lagarias
; Sandor Szabo
; | | Date: |
16 Aug 2000 | | Journal: | J. Fourier Anal. Appl. 7 (2001), 63--70. | | Subject: | Functional Analysis; Spectral Theory MSC-class: Primary: 47A13 Secondary: 11K70, 42B05 | math.FA math.SP | | Abstract: | 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. | | Source: | arXiv, math.FA/0008132 | | Services: | Forum | Review | PDF | Favorites | |
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