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)
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