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)