Substitution tilings and separated nets with similarities to the integer lattice Academic Article uri icon

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 ∥Φ(y) − y∥ < ∞. 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.

publication date

  • January 1, 2011