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On characterizations of spectra and tilings. (English) Zbl 1106.42032

Author’s abstract: In a recent paper, J. C. Lagarias, J. A. Reeds and Y. Wang [Duke Math. J. 103, No. 1, 25–37 (2000; Zbl 0978.42007)] established a characterization of spectra and tilings that can be used to prove a conjecture of P. E. T. Jorgensen and S. Pedersen [J. Fourier Anal. Appl. 5, No. 4, 285–302 (1999; Zbl 1050.42016)] by Keller’s criterion (1930). Different techniques to prove these facts have also been developed by by the reviewer and A. Iosevich and S. Pedersen [Int. Math. Res. Not. 1998, No. 16, 819–828 (1998; Zbl 0926.52020)]. The primary aim of this paper is to present an elementary method of describing certain characterizations of spectra and tilings. To illustrate this method, we first give a simple proof of this characterization. We then use the method to derive some characteristic results connected with the dual Fuglede’s spectral-set conjecture. The results here extend several known conclusions in a simple manner.

MSC:

42C99 Nontrigonometric harmonic analysis
11H31 Lattice packing and covering (number-theoretic aspects)
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
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