Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Cut-and-Project Schemes for Pisot Family Substitution Tilings

Version 1 : Received: 19 September 2018 / Approved: 19 September 2018 / Online: 19 September 2018 (07:44:07 CEST)

A peer-reviewed article of this Preprint also exists.

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

Journal reference: Symmetry 2018, 10, 511
DOI: 10.3390/sym10100511

## Abstract

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].

## Keywords

Pisot substitution tilings; pure point spectrum; regular model set; algebraic coincidence

## Subject

MATHEMATICS & COMPUTER SCIENCE, Geometry & Topology

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