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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