The Net Created from the Penrose Tiling Is Bilipschitz to the Integer Lattice

Let τ1 be the Penrose tiling on R 2 [G77] (using kites and darts) with edges of lengths 1 and φ, where φ is the golden ratio 1+ √ 5 2 . Let U be a square of the form [j1, j1 + l] × [j2, j2 + l] where l = 2, i, j1, j2 ∈ N. Notice that when we have a tiling of the plane (so that the diameter of its tiles is bounded) we can produce a separated net from it. We do it by placing one point in each tile, and remember to maintain the minimal distance property from (1.1). Obviously, by placing the points differently in the tiles we will get different separated nets, but all these nets will be in the same biLipschitz equivalence class. Thus we can say that every tiling creates a separated net. Let Y be the separated net which is created by the tiling τ1. For a number ρ > 0 we define as in [BK02] (1)