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 UveZN(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 Pu 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 Tu.
The following notation and technical lemma makes easier some calcu-
lations in future sections.
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