Lee, J.-Y.; Akiyama, S.; Nagai, Y. Cut-and-Project Schemes for Pisot Family Substitution Tilings. Symmetry2018, 10, 511.
Lee, J.-Y.; Akiyama, S.; Nagai, Y. Cut-and-Project Schemes for Pisot Family Substitution Tilings. Symmetry 2018, 10, 511.
We consider Pisot family substitution tilings in $\R^d$ whose dynamical spectrum is pure point. There are two cut-and-project schemes(CPS) which arise naturally: one from the Pisot family property and the other from the pure point spectrum respectively. The first CPS has an internal space $\R^m$ for some integer $m \in \N$ defined from the Pisot family property, and the second CPS has an internal space $H$ which is an abstract space defined from the property of the pure point spectrum. However it is not known how these two CPS's are related. Here we provide a sufficient condition to make a connection between the two CPS's. In the case of Pisot unimodular substitution tiling in $\R$, the two CPS's turn out to be same due to [5, Remark 18.5].
Pisot substitution tilings; pure point spectrum; regular model set; algebraic coincidence
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