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

Abstract

A separated net is a set of points which is relatively dense and uniformly discrete (another name for a Delone set). We are dealing with tilings and separated nets in Euclidean spaces and with the question whether a given separated net is biLipschitz to the integer lattice. In this paper we show, as an answer to a question of Burago and Kleiner, that the net that is obtained form the Penrose tiling is biLipschitz to the integer lattice.