Substitution Tilings and Separated Nets with Similarities to the Integer Lattice

2008/10/29 by Yaar Solomon, Solomon, Yaar · 1 citation
Computer Science · Materials Science · Mathematics · #Advanced Combinatorial Mathematics #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG) #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.0810.5225

openalex publication_date 2008/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• The net created from the Penrose tiling is biLipschitz to the integer lattice

Cited by

• The Gauss circle problem for Penrose tilings

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