We consider Pisot family substitution tilings in
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
for some integer
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
, the two CPSs turn out to be same due to the remark by Barge-Kwapisz.
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