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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)
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