Substitution Tilings and Separated Nets with Similarities to the Integer Lattice

Details Substitution Tilings and Separated Nets with Similarities to the Integer Lattice Substitution Tilings and Separated Nets with Similarities to the Integer Lattice

2008-10-29

arxiv.org/abs/0810.5225v2

Mathematics

Metric Geometry

Scientific paper

We show that any primitive substitution tiling of the plane creates a separated net which is biLipschitz to the integer lattice. Then we show that if H is a primitive Pisot substitution in an Euclidean space, for every separated net Y, that corresponds to some tiling of the tiling space, there exists a bijection F between Y and the integer lattice that translate every element of Y a bounded distance. As a corollary we get that we have such an F for any separated net that corresponds to a Penrose Tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.

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