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Substitution tilings and separated nets with similarities to the integer lattice. (English) Zbl 1217.52012
Summary: We show that any primitive substitution tiling of $$\mathbb R^2$$ creates a separated net which is biLipschitz to $$\mathbb Z^{2}$$. Then we show that if $$H$$ is a primitive Pisot substitution in $$\mathbb R^d$$, for every separated net $$Y$$, that corresponds to some tiling $$\tau \in X_H$$, there exists a bijection $$\Phi$$ between $$Y$$ and the integer lattice such that $$\sup_{y\in Y}\|\Phi (y)-y\|<\infty$$. As a corollary, we get that we have such a $$\Phi$$ for any separated net that corresponds to a Penrose tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.

##### MSC:
 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry) 05B45 Combinatorial aspects of tessellation and tiling problems 37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) 52C20 Tilings in $$2$$ dimensions (aspects of discrete geometry)
##### Keywords:
separated net; Pisot substitution; Penrose tiling
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##### References:
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