Substitution Tilings and Separated Nets with Similarities to the Integer Lattice

TL;DR

This paper demonstrates that certain substitution tilings in the plane and Euclidean spaces produce separated nets that are biLipschitz equivalent to the integer lattice, with explicit bijections translating points within bounded distances.

Contribution

It establishes biLipschitz equivalence between separated nets from primitive substitution tilings and the integer lattice, extending to Penrose tilings and related structures.

Findings

Separated nets from primitive substitution tilings are biLipschitz to the integer lattice.

Explicit bijections exist translating nets to the lattice within bounded distances.

Results apply to Penrose tilings and general primitive Pisot substitutions.

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.