12 A. FORREST, J. HUNTON AND J. KELLENDONK A point v G K^ is said to be non-singular if the boundary, 9£, of E does not intersect ZN + v. We write NS for the set of non-singular points in M.N. These points are also called regular in the literature. Let i \ = S n (ZN + v), the strip point pattern. Define Pv = TT(PV), a subset of E called the projection point pattern. In what follows we assume E and K are fixed and suppress mention of them as a subscript or argument. Lemma 2.2 With the notation above, i/ NS is a dense G$ subset ofM.N invariant under translation by E. ii/ If u G NS, then NS n (Q + u) is dense in Q + u. Hi/ If u G NS and F is a vector subspace of RN complementary to E, then NS D (Q + u) n F is dense in (Q + u) n F. Proof i/ Note that RN \ NS is a translate of the set U veZN (dK + E + v) (where the boundary is taken in E^) and our conditions on K complete the proof. ii/ NS n (Q + u) D E + ZN + u. hi/ ( £ + Z ^ + u ) f l F = (Q + u) n F. D Remark 2.3 The condition on the acceptance domain K is a topo- logical version of the condition of [H]. We note that our conditions include the examples of acceptance domains with fractal boundaries which have recently interested quasicrystalographers [BKS] [Sm] [Z] [GLJJ]. In the original construction [dBl] [KD] K = TT^QO, 1 ] N ) . We call this the canonical acceptance domain and we reserve the name canonical projection method pattern for the patterns P u produced from this acceptance domain. Sometimes this is shortened to canonical pattern for convenience. This is closely related to the canonical projection tiling, defined by [OKD] formed by a canonical acceptance domain, u G NS and projecting onto E those d-dimensional faces of the lattice ZN + u which are contained entirely in E. We write this tiling T u . The following notation and technical lemma makes easier some calcu- lations in future sections.
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