Abstract
We show that the spectral-tile implication in the Fuglede conjecture in dimension 1 is equivalent to a Universal Tiling Conjecture and also to similar forms of the same implication for some simpler sets, such as unions of intervals with rational or integer endpoints.
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Acknowledgements
This work was done while the first named author (P.J.) was visiting the University of Central Florida. We are grateful to the UCF-Math Department for hospitality and support. The authors are pleased to thank Professors Deguang Han, Steen Pedersen, Qiyu Sun and Feng Tian for helpful conversations. P.J. was supported in part by the National Science Foundation, via a Univ of Iowa VIGRE grant. We thank the anonymous referee for the carefully reading the manuscript and for his/her suggestions that improved the paper significantly. This work was partially supported by a grant from the Simons Foundation (#228539 to Dorin Dutkay).
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Communicated by Karlheinz Gröchenig.
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Dutkay, D.E., Jorgensen, P.E.T. On the Universal Tiling Conjecture in Dimension One. J Fourier Anal Appl 19, 467–477 (2013). https://doi.org/10.1007/s00041-013-9264-7
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DOI: https://doi.org/10.1007/s00041-013-9264-7