14 A. FORREST, J. HUNTON AND J. KELLENDONK Note that we speak of tilings and yet only consider point patterns. In both examples, the projection tiling [OKD] is conjugate to both the corresponding strip point pattern and projection point pattern, a fact proved in greater generality in section 8. We develop these geometric ideas in the following lemmas. The next is Theorem 2.3 from [Se]. Theore m 2.8 Suppose that ZN is in standard position in WN and suppose that (/): 1$LN — W1 is a surjective linear map. Then there is a direct sum decomposition R n = V © W into real vector subspaces such that (f)(ZN) nV is dense in V, (j)(ZN) nW is discrete and (j)(ZN) = (v n /(zN)) + (wn (j){zN)). a We proceed with the following refinement of Proposition 2.15 of [Se]. Lemm a 2.9 Suppose that ZN is in standard position in WN and sup- pose that (j):M.N — F is an orthogonal projection onto F a sub- space ofM.N. With the decomposition of F implied by Theorem 2.8, (F n ZN) + (V n j(ZN)) C /(ZN) as a finite index subgroup. Also, the lattice dimension of F n ZN equals dimF — dim V and the real vector subspace generated by F n 7LN is orthogonal to V. Proof Suppose that U is the real linear span of A = FnZN. Since A is discrete, the lattice dimension of A equals the real space dimension oiU. The argument of the proof of Proposition 2.15 in [Se] shows that each element of F D ZN is orthogonal to V. Therefore we have dim^([7) dim^(F) — dimM(V) immediately. Consider the rational vector space QN, contained in RN and con- taining Z ^ , both in canonical position. Let U' be the rational span of A and note that Uf = UnQN and that dimQ(/7/) = dimM(£7). Let Uf± be the orthocomplement of Uf with respect to the standard inner product in Q ^ so that, by simple rational vector space arguments, Q ^ = U'®U,JL. Thus (UfnZN) + (Uf±nZN) forms a discrete lattice of dimension N. Extending to the real span, we deduce that (UP\ZN) + (U1- flZ^) is a discrete sublattice of ZN of dimension iV, hence a subgroup of finite index. Also the lattice dimension of U n ZN and U1- D ZN are equal to dim^(t/) and dim^(f7 ± ) respectively. Let L — (U1- n ZN) be considered as a sublattice of U^. It is integral (with respect to the restriction of the inner product on M,N)

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2002 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.