Cut-and-Project Schemes for Pisot Family Substitution Tilings

We consider Pisot family substitution tilings in ${\mathbb{R}}^d$ 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 ${\mathbb{R}}^m$ for some integer $m\in \mathbb{N}$ 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 $\mathbb{R}$ , the two CPSs turn out to be same due to the remark by Barge-Kwapisz. View Full-Text