Substitution tilings and separated nets with similarities to the integer lattice

Yaar Solomon

doi:10.1007/s11856-011-0018-4

Substitution tilings and separated nets with similarities to the integer lattice

Yaar Solomon

Department of Mathematics

Ben-Gurion University of the Negev

Research output: Contribution to journal › Article › peer-review

Abstract

We show that any primitive substitution tiling of ℝ² creates a separated net which is biLipschitz to ℤ². Then we show that if H is a primitive Pisot substitution in ℝ^d, for every separated net Y, that corresponds to some tiling τ ∈ X_H, there exists a bijection Φ between Y and the integer lattice such that sup_y∈Y{double pipe}Φ(y) - y{double pipe} < ∞. As a corollary, we get that we have such a Φ for any separated net that corresponds to a Penrose Tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.

Original language	American English
Pages (from-to)	445-460
Number of pages	16
Journal	Israel Journal of Mathematics
Volume	181
Issue number	1
DOIs	https://doi.org/10.1007/s11856-011-0018-4
State	Published - 1 Mar 2011

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s11856-011-0018-4

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title = "Substitution tilings and separated nets with similarities to the integer lattice",

abstract = "We show that any primitive substitution tiling of ℝ2 creates a separated net which is biLipschitz to ℤ2. Then we show that if H is a primitive Pisot substitution in ℝd, for every separated net Y, that corresponds to some tiling τ ∈ XH, there exists a bijection Φ between Y and the integer lattice such that supy∈Y\{double pipe\}Φ(y) - y\{double pipe\} < ∞. As a corollary, we get that we have such a Φ for any separated net that corresponds to a Penrose Tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.",

author = "Yaar Solomon",

note = "Funding Information: Acknowledgements. This research was supported by the Israel Science Foundation. This work is a part of the author{\textquoteright}s Master thesis under the supervision of Barak Weiss whose endless support and guidance are deeply appreciated. The author also wishes to thank Bruce Kleiner for suggesting the problem about the Penrose Tiling. After [S08] appeared on the web it was brought to the author{\textquoteright}s attention that there is another paper, [DSS95], where a sketch of a proof for Corollary 1.5 is given.",

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doi = "10.1007/s11856-011-0018-4",

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AU - Solomon, Yaar

N1 - Funding Information: Acknowledgements. This research was supported by the Israel Science Foundation. This work is a part of the author’s Master thesis under the supervision of Barak Weiss whose endless support and guidance are deeply appreciated. The author also wishes to thank Bruce Kleiner for suggesting the problem about the Penrose Tiling. After [S08] appeared on the web it was brought to the author’s attention that there is another paper, [DSS95], where a sketch of a proof for Corollary 1.5 is given.

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AB - We show that any primitive substitution tiling of ℝ2 creates a separated net which is biLipschitz to ℤ2. Then we show that if H is a primitive Pisot substitution in ℝd, for every separated net Y, that corresponds to some tiling τ ∈ XH, there exists a bijection Φ between Y and the integer lattice such that supy∈Y{double pipe}Φ(y) - y{double pipe} < ∞. As a corollary, we get that we have such a Φ 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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Substitution tilings and separated nets with similarities to the integer lattice

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