Substitution tilings and separated nets with similarities to the integer lattice

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23 Scopus citations

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.

Original languageEnglish
Pages (from-to)445-460
Number of pages16
JournalIsrael Journal of Mathematics
Volume181
Issue number1
DOIs
StatePublished - 1 Mar 2011

ASJC Scopus subject areas

  • Mathematics (all)

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