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Cut-and-Project Schemes for Pisot Family Substitution Tilings

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Department of Mathematics Education, Catholic Kwandong University, Gangneung, Gangwon 210-701, Korea, or KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Korea
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Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan
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Lehrstuhl für Mathematik und Statistik, Department Mathematik und Informationstechnologie, Montanuniversität, Franz Josef Strasse 18, A-8700 Leoben, Austria
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Author to whom correspondence should be addressed.
Symmetry 2018, 10(10), 511; https://doi.org/10.3390/sym10100511
Received: 19 September 2018 / Revised: 9 October 2018 / Accepted: 11 October 2018 / Published: 16 October 2018
We consider Pisot family substitution tilings in R d whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space R m for some integer m N defined from the Pisot family property, and the second CPS has an internal space H that is an abstract space defined from the condition of the pure point spectrum. However, it is not known how these two CPSs are related. Here we provide a sufficient condition to make a connection between the two CPSs. For Pisot unimodular substitution tiling in R , the two CPSs turn out to be same due to the remark by Barge-Kwapisz. View Full-Text
Keywords: Pisot substitution tilings; pure point spectrum; regular model set; algebraic coincidence Pisot substitution tilings; pure point spectrum; regular model set; algebraic coincidence
MDPI and ACS Style

Lee, J.-Y.; Akiyama, S.; Nagai, Y. Cut-and-Project Schemes for Pisot Family Substitution Tilings. Symmetry 2018, 10, 511.

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