Uniqueness in Discrete Tomography of Planar Model Sets

0.7 MB·English

by  Christian Huck

UNIQUENESS IN DISCRETE TOMOGRAPHY
OF PLANAR MODEL SETS
7
0 CHRISTIANHUCK
0
2
Abstra
t. The problem of dΛetermining (cid:28)nite subsets of
hara
teristi
pXlanar model sets
n
(mathemati
alXquasi
rystals) ,
ualled
y
lotomi
model sets, by parallel -rays is
onsid-
a
J ered. Here,an -rayindire
tion uofa(cid:28)nitesubsetoftheplaneXgivestheΛnumberofpoints
5 in the set on ea
h line parallel to . For pra
ti
al reasons, oΛnl−yΛ-rays in -dire
tions, i.e.,
dire
tionsparallel tonon-zeroelementsofthedi(cid:27)eren
eset ,arepermitted. Inparti
-

] ular, by
ombining methods from algebrai
nuΛmber theory and C
on⊂veΛxity, it is shown that
G
the
onvex subsets of a
Λy
lotomi
model set , i.e., (cid:28)nite sets wΛhose
onveXx hulls
M
ontainnonew poΛintsof ,are determined,amongall
Λonvexsubsetsof , bytheir -rays
infourpres
ribed -dire
tions,whereasanysetofthree -dire
tionsdoesnotsu(cid:30)
eforthis
.
h purpose. We also study the intera
tive te
hniqueof su

essive deXtermination in the
ase of
at
y
lotomi
model sets, in wXhi
h the information from previous -rays is used in de
iding
m the dire
tion for the nΛext -ray. In parti
ular, it is shown thΛat the (cid:28)nite subsets of any

y
lotomi
model set
an be su

essively determined by two -dire
tions. All results are
[
illustratedbymeansofwell-knownexamples,i.e., the
y
lotomi
modelsetsasso
iated with

2 the square tiling, the triangle tiling, the tiling of Ammann-Beenker, the Tübingen triangle
v tiling and theshield tiling.
1
4
1
1
0
7 1. Introdu
tion
0
/ Dis
rete tomography is
on
erned with the inverse problem of retrieving information about
h
somedis
reteobje
tfrom(generallynoisy)informationaboutitsin
iden
eswith
ertainquery
t
a
sets. A typi
al example is the re
onstru
tion of a (cid:28)nite point set from its line sums in a small
m
number of dire
tions. More pre
isely, a (dis
rete parallel) X-ray of a (cid:28)nite subset of Eu
lidean
: d d u d u
v -spa
e R indire
tion gives the number of pointsin theset onea
h linein R parallel to .
i
X Inthe
la2ssi
alsetting,motivatedby
rystals,thepLositionsdtobededterm2inedliveonthesquSare
r latdti
e Z or, more generally, on aGrbitraryd latti
es in R , whereS ≥S.:=(Hesre, assusb,sset Sof
a ′ ′
R is said to live on a subgroup of R when its di(cid:27)eren
e set − { − | ∈ }
G
is a subset of .) In fa
t, many of the problems in dis
rete tomography have been studied
2
on Z , the
lassi
al planar setting of dis
rete tomography; see [24℄, [23℄ and [19℄. In the
longer run, by also having other stru
tures than perfe
t
rystals in mind, one has to take
into a

ount wider
lasses of sets, or at least signi(cid:28)
ant deviations from the latti
e stru
ture.
As an intermediate step between periodi
and random (or amorphous) Delone sets (de(cid:28)ned
below), we
onsider systems of aperiodi
order, more pre
isely, of so-
alled model sets (or
mathemati
al quasi
rystals), whi
h are
ommonly a

epted as a good mathemati
al model for
quasi
rystalline stru
tures in nature [40℄. It is interesting to know whether these sets behave
like latti
es as far as dis
rete tomography is
on
erned.
The main motivation for our interest in the dis
rete tomography of model sets
omes from
the physi
al existen
e of quasi
rystals that
an be des
ribed as aperiodi
model sets together
1
2 C.HUCK
withthedemandofmaterialss
ien
etore
onstru
tthree-dimensional(quasi)
rystalsorplanar
layers of them from their images under quantitative high resolution transmission ele
tron
mi
ros
opy (HRTEM) in a small number of dire
tions. In fa
t, in [26, 37℄ a te
hnique is
des
ribed, based on HRTEM, whi
h
an e(cid:27)e
tively measure the number of atoms lying on
lines parallel to
ertain dire
tions; it is
alled QUANTITEM (quantitative analysis of the
information
oming from transmission ele
tron mi
ros
opy). At present, the measurement
of the number of atoms lying on a line
an only be a
hieved for some
rystals; see [26, 37℄.
However, it is reasonable to expe
t that future developments in te
hnology will improve this
situation.
Whether or not one has future appli
ations in materials s
ien
e in mind, the starting point
willalwaysbeaspe
i(cid:28)
stru
turemodel. Thismeansthatthespe
i(cid:28)
typeofthe(quasi)
rystal
X
isgiven (see[2, 38℄for the
on
ept), andone is
onfronted withthe -ray data of anunknown
d
(cid:28)nite subset of it. This is the
orre
t analogue of starting with translates of Z in the
lassi
al
setting.
Here, our investigation of the dis
rete tomography of systems with aperiodi
order will
be restri
ted to the study of two-dimensional systems. Fortunately, solving the problems of
dis
rete tomography for two-dimensional systems with aperiodi
order also lies at the heart
of solving the
orresponding problems in three dimensions. This is due to the fa
t that model
sets possess, as it is the
ase for latti
es, a dimensional hierar
hy, meaning that any model set
d d 1
in dimensions
an be sli
ed into model sets of dimension − .
More pre
isely, we
onsider a well-known
lass of planar model sets whi
h
an be des
ribed
in algebrai
terms. It will be a ni
e property of this
lassof planar model setsthat it
ontains,
2
asaspe
ial
ase,arbitrarytranslatesofthesquarelatti
eZ ,whi
h
orrespondstothe
lassi
al
planar setting of dis
rete tomography.
Let us be
ome more spe
i(cid:28)
. By using the Minkowski representation of algebrai
number
n / 1,2 Λ =
(cid:28)elds, we introdu
e, for ∈ { }, the
orresponding
lass of
y
lotomi
model sets ⊂ C∼
2 [ζ ] ζ n ζ = e2πi/n
n n n
R whi
hliveonZ ,where isaprimitive throotofunityinC,e.g., . TheZ-
[ζ ] n (ζ ) n / 1,2,3,4,6
n n
moduleZ istheringofintegersinthe th
y
lotomi
(cid:28)eldQ ,and,for ∈ { },
Λ
when viewed as a subset of the plane, is dense, whereas
y
lotomi
model sets are Delone
sets, i.e., they are uniformly dis
rete and relatively dense. In fa
t, model sets are even Meyer
Λ Λ
sets, meaning that also − is uniformly dis
rete; see [29℄. It turns out that, ex
ept the
[ζ ] n 3,4,6
n

y
lotomi
model sets living on Z with ∈ { } (these are exa
tly the translations
Λ
of the square and the triangular latti
e, respe
tively),
y
lotomi
model sets are aperiodi
,
meaning that they have no translational symmetries at all. Well-known examples are the
N
planar model sets with -fold
y
li
symmetry that are asso
iated with the square tiling
n = N = 4 2n = N = 6 n = N = 8
( ), the triangle tiling ( ), the Ammann-Beenker tiling ( ),
2n = N = 10 n = N = 12
the Tübingen triangle tiling ( ) and the shield tiling ( ). Note that
5,8,10 12
orders and o

ur as standard
y
li
symmetries of genuine quasi
rystals [40℄, whi
h
are basi
ally sta
ks of planar aperiodi
layers.
In general, the re
onstru
tion problem of dis
rete tomography
an possess rather di(cid:27)erent
solutions. This leads to the investigation of the
orresponding uniqueness problem, i.e., the
Λ X
(unique) determinationof (cid:28)nite subsetsof a(cid:28)xed
y
lotomi
model set by -rays ina small
X Λ
number of suitably pres
ribed dire
tions. For pra
ti
al reasons, only -rays in -dire
tions,
Λ Λ Λ
i.e., dire
tions parallel to non-zero elements of the di(cid:27)eren
e set − of , are permitted.
DISCRETE TOMOGRAPHY 3
Λ
We say that a subset E of the set of all (cid:28)nite subsets of a (cid:28)xed
y
lotomi
model set is
X U F F
′
determined by the -rays in a (cid:28)nite set of dire
tions if di(cid:27)erent sets and in E have
X u U X F F
′
di(cid:27)erent -rays, i.e., ifthereisadire
tion ∈ su
hthatthe -raysof and indire
tion
u
di(cid:27)er.
Λ
Without the restri
tion to -dire
tions, one
an prove that the (cid:28)nite subsets of a (cid:28)xed
Λ X

y
lotomi
model set
an be determined by one -ray (Proposition 7). In fa
t, it turns
X Λ
out that any -ray in a non- -dire
tion is suitable for this purpose. However, in pra
ti
e
X Λ
(HRTEM), -rays in non- -dire
tions are meaningless sin
e the resolution
oming from su
h
X
-rays would not be good enough to allow a quantitative analysis (cid:21) neighbouring lines are
not su(cid:30)
iently separated.
F
Proposition 9 demonstrates that the (cid:28)nite sets of
ardinality less than or equal to some
k Λ k+1 X
∈ N in a (cid:28)xed
y
lotomi
model set are determined by any set of -rays in pairwise
Λ
non-parallel -dire
tions. However, in pra
ti
e, one is interested in the determination of (cid:28)nite
X 3 5
sets by -rays in a small number of dire
tions sin
e after about to images taken by
HRTEM, the obje
t may be damaged or even destroyed by the radiation energy. Observing
106 109
that the typi
al atomi
stru
tures to be determined
omprise about to atoms, one
realizes that the last result is not pra
ti
al at all. In fa
t, it is also shown that any (cid:28)xed
X Λ
number of -rays in -dire
tions is insu(cid:30)
ient to determine the entire
lass of (cid:28)nite subsets
Λ
of a (cid:28)xed
y
lotomi
model set (Proposition 8).
In view of this, it is ne
essary to impose some restri
tion in order to obtain positive unique-
ness results.
Λ
As a (cid:28)rst option, we restri
t the set of (cid:28)nite subsets of a (cid:28)xed
y
lotomi
model set
R > 0 Λ
under
onsideration, i.e., we
onsider for every the
lass of bounded subsets of with
R Λ Λ
diameter less than . Sin
e is uniformly dis
rete, bounded subsets of are (cid:28)nite. It is
R > 0 Λ
shown that, for all and for all
y
lotomi
model sets , there are two non-parallel
Λ Λ R
pres
ribed -dire
tions su
h that the bounded subsets of with diameter less than are
Λ X
determined, among all su
h subsets of , by their -rays in these dire
tions (Corollary 11).
In fa
t, in Theorem 13, it is shown that the
orresponding result holds for arbitrary Delone
Λ d d 2 Λ Λ
sets ⊂ R , ≥ , of (cid:28)nite lo
al
omplexity, the latter meaning that the di(cid:27)eren
e set −
L d d 2
is
losed and dis
rete. Thereby, also the
ase of translates of latti
es in R , ≥ , is
in
luded (Corollary 12). Moreover, even the
orresponding results for orthogonal proje
tions
1 (d 1)
on orthogonal
omplements of -dimensional (respe
tively, − -dimensional) subspa
es
Λ Λ L L =L
generated by elements of − (resp., − ) hold; the multipli
ity information
oming
X
along with -rays is not needed. Unfortunately, these results are limited in pra
ti
e be
ause,
in general, one
annot make sure that all the dire
tions whi
h are used yield images of high
enough resolution.
Λ
As a se
ond option, we restri
t the set of (cid:28)nite subsets of a (cid:28)xed
y
lotomi
model set
Λ C Λ
by
onsidering the
lass of
onvex subsets of . They are (cid:28)nite sets ⊂ whose
onvex hulls
Λ C Λ C = conv(C) Λ

ontain no new points of , i.e., (cid:28)nite sets ⊂ with ∩ . In Theorem 15, it is
Λ
shown that there are four pairwise non-parallel pres
ribed -dire
tions su
h that the
onvex
Λ [ζ ]
n
subsets of any
y
lotomi
model set living on Z are determined, among all
onvex
Λ X
subsets of , by their -rays in these dire
tions. It is further shown that three pairwise non-
Λ
parallel -dire
tions never su(cid:30)
e for this purpose, whereas any seven pairwise non-parallel
Λ
-dire
tions whi
h satisfy a
ertain relation (see below for details) have the property that the
4 C.HUCK
Λ

onvex subsets of any
y
lotomi
model set are determined, among all
onvex subsets of
Λ X
, by their -rays in these dire
tions. It is also demonstrated that, in a sense, the number
seven
annot be redu
ed to six in the latter result. These results heavily depend on a massive
Λ
restri
tion of the set of -dire
tions. We shall also remove this restri
tion and present expli
it
Λ
results in the
ase of the important sub
lass of
y
lotomi
model sets with
o-dimension
2 [ζ ] n 5,8,10,12 p
n
, i.e.,
y
lotomi
model sets living on Z , where ∈ { }. Here, by using -adi
valuations, weareabletopresentamoresuitableanalysiswhi
h showsthatuniquenesswillbe
Λ
provided byanysetoffour -dire
tionswhoseslopes(suita(bζly+orζ¯de)/red)yielda
rossratiothat
n n
doesnotmap underthe (cid:28)eldnorm of the(cid:28)eld extension Q Q to a
ertain (cid:28)nite set of
rational numbers (Theorem 16)(.ζ(I+t wζ¯il)l turn ount that these
ross ratio(ζs a)re indeed elements
n n n
of the maximal real sub(cid:28)eld Q of the th
y
lotomi
(cid:28)eld Q .) It is also shown
that, by the same analysis, similar results
an be obtained for arbitrary
y
lotomi
model sets
2
(Theorem 17). In the
ase of
y
lotomi
model sets with
o-dimension , we shall be able
Λ
to present sets of four -dire
tions whi
h provide uniqueness and yield images of rather high
resolutioninthequantitative HRTEMofthe
orresponding (aperiodi
)
y
lotomi
modelsets,
the latter making these results look promising.
U
A major task in a
hieving the above results involves examining so-
alled -polygons in
y-
Λ U

lotomi
model sets , whi
h exhibit a weak sort of regularity. In the
ontext of -polygons,
the question for a(cid:30)nely regular polygons in the plane with all their verti
es in a (cid:28)xed
y-
Λ

lotomi
model set arises rather naturally. Chrestenson [12℄ has shown that any (planar)
d d 2 3,4 6
regular polygon whose verti
es are
ontained in Z for some ≥ must have or verti
es.
More generally, Gardner and Gritzmann [17℄ have
hara
terized the a(cid:30)nely regular latti
e
polygons, i.e., images of non-degenerate regular polygons under a non-singular a(cid:30)ne transfor-
2 2 2
mation R −→ R whose verti
es are
ontained in Z . It turned out that these are pre
isely
the a(cid:30)nely regular triangles, parallelograms and hexagons. Clearly, their result remains valid
2 [ζ ]
3
when one repla
es the square latti
e Z by the triangular latti
e Z , sin
e the latter is the
image of the (cid:28)rst under an invertible linear transformation of the plane. Observing that both
[ζ ]
n
the square and triangular latti
e are examples of rings of integers Z in
y
lotomi
(cid:28)elds
(ζ )
n
Q , one
an ask for a generalization of the above result to all these obje
ts, viewed as
Z-modules in the plane. Using standard results from algebra and algebrai
number theory, we
provide a
hara
terization in terms of a simple divisibility
ondition (Theorem 6). Moreover,
we show that our
hara
terization of a(cid:30)nely regular polygons remains valid when restri
ted
Λ
to the
orresponding
lass of
y
lotomi
model sets (Corollary 7).
As a third option, we
onsider the intera
tive te
hnique of su

essive determination, in-
X
trodu
ed by Edelsbrunner and Skiena [14℄ for
ontinuous -rays, in the
ase of aperiodi
X

y
lotomi
model sets, in whi
h the information from previous -rays may be used in de
id-
X
ingthe dire
tion for the next -ray. Here, itisshown thatthe (cid:28)nite subsetsof any
y
lotomi
Λ X Λ
model set
an be su

essively determined by -rays in two non-parallel -dire
tions. In
n / 1,2 [ζ ]
n
fa
t, it is shown that, for ∈ { }, the (cid:28)nite subsets of the ring of
y
lotomi
integers Z
X [ζ ]
n

an be su

essively determined by -rays in two non-parallel Z -dire
tions, i.e., dire
tions
[ζ ]
n
parallel to a non-zero element of Z (Theorem 20). Again, the
orresponding results do
1
even hold for orthogonal proje
tions on orthogonal
omplements of -dimensional subspa
es
Λ Λ X
generated by elements of − ; the multipli
ity information
oming alsong with -rays is
DISCRETE TOMOGRAPHY 5
again not needed. Unfortunately, these results are again limited in pra
ti
e be
ause, in gen-
eral, one
annot make sure that all the dire
tions whi
h are used yield images of high enough
resolution.
Previous studies have fo
ussed on the `an
hored'
ase that the underlying ground set is
X
lo
ated in a linear spa
e, i.e., in a spa
e with a spe
i(cid:28)ed lo
ation of the origin. The -ray
data is then taken with respe
t to this lo
alization. Note that the rotational orientation of
a (quasi)
rystalline probe in an ele
tron mi
ros
ope
an rather easily be done in the di(cid:27)ra
-
tion mode, prior to taking images in the high-resolution mode, though, in general, a natural

hoi
e of a translational origin is not possible. This problem has its origin in the pra
ti
e of
X
quantitative HRTEM sin
e, in general, the -ray information does not allow us to lo
ate the
examined sets. We believe that, in order to obtain appli
able results, one has to deal with
this `non-an
hored'
ase.
In dis
ussing inverse problems, it is important to distinguish between determination and
Λ
re
onstru
tion. The problem of re
onstru
ting (cid:28)nite subsets of
y
lotomi
model sets given
X Λ
-rays in two -dire
tions is treated in [4℄. There, it is shown that, for a large
lass of

y
lotomi
model sets, this problem
an be solved in polynomial time in the real RAM-model
of
omputation. For
orresponding results in the latti
e
ase, see [19℄.
Note that this work provides a generalization of well-known results, obtained by Gardner
L d d 2
and Gritzmann [17℄ for the `an
hored'
ase of (cid:28)xed translates of latti
es in R , ≥ ,
to the
lass of
y
lotomi
model sets. Additionally, we shall present results whi
h deal with
the `non-an
hored'
ase des
ribed above. Several of our proofs are only a slight modi(cid:28)
ation
of the proofs from [17℄. The key role in this text is played by Lemma 21, whi
h rests on a
p
result from the theory of Pisot-Vijayaraghavan numbers [34℄, and -adi
valuations. For more
details, other settings, and the histori
al ba
kground of the above uniqueness problems, we
would like to refer the reader to [17℄ and referen
es therein.
2. Algebrai
Ba
kground, Definitions and Notation
= 1,2,3,...
Natural numbers are always assumed to be positive, i.e., N { }. Throughout
the text, wde use the
donvention that the symbol ⊂ idn
ludes equality.SdW1e denote the norm in
−
Eu
lidean -spa
e R by k·k. The unit sphere in R is denoted by , i.e.,
Sd 1 = x d x = 1 .
−
∈R k k
n (cid:12) o
Sd 1 (cid:12) x x
− (cid:12)
Moreover, the elements of are also
alled dire
tions. If ∈ R, then ⌊ ⌋ denotes the
x r > 0 x d B (x)
r
greatest integer less than or equal to . For and ∈ R , is the open ball of
r x k,l (k,l) [k,l]
radius about . If ∈ N, then and denote their greatest
ommon divisor and
least
ommon multiple, respe
tively. There should be no
onfusion with the usual notation of
S d k R > 0
open (resp.,
losed) and bounded real intervals. For a subset ⊂ R , ∈ N and , we
card(S) (S) (S) (S) S S ∂S S S S conv(S) diam(S)
k <R ◦
denote by , F , F≤ , D , , , , h iZ, h iQ, h iR, ,
S
S
and 1 the
ardinality, the setof (cid:28)nite subsets, the setof (cid:28)nite subsets of having
ardinality
k S R
less than or equal to , the set of subsets of with diameter less than , interior,
losure,
boundary, Z-linear hull, Q-linear hull, R-linear hull,
onvex hull, diameter and
hara
teristi
S
fun
tion of , respe
tively. The notation of the
losure operator should not be
onfused with
S
the usual notation of
omplex
onjugation. The dimension of is the dimension of its a(cid:30)ne
6 C.HUCK
aff(S) dim(S) T d S
hull , and is denoted by . Further, a linear subspa
e of R is
alled an -
S S := s s s,s S
′ ′
subspa
e iuf it Sisdg1enerated by elSements of the di(cid:27)eren
e set − { − | S∈ }S. A
−
dire
tion ∈ is
alled an -dire
tion if it is parallel to a non-zero element of − . If
T d x d T
is a linear subspa
e of R , we denote the orthogonal proje
tion of an element of R on
xT S d T S T
by | . Moreover, we denote the orthogonal proje
tion of a subset of R on by | . The
T T
⊥
orthogonal
omplement of is denoted by . If we use this notation with respe
t to other
d
than the
anoni
al inner produ
t on R , we shall say so expli
itly. The symmetri
di(cid:27)eren
e
A B A B := (A B) (B A) S d
of two sets and is de(cid:28)ned as △ \ ∪ \ . A subset of R is said to
G d S S G
live on a subgroup of R when its di(cid:27)eren
e set − is a subset of . Obviously, this is
t d S t+G
equivalent to the existen
e of a suitable ∈ R su
h that ⊂ . Finally, the
entroid of
F ( d) ( f)/card(F)
an element ∈ F R is de(cid:28)ned as f∈F .
d F P ( d) u Sd 1
−
De(cid:28)nition 1. Let ∈ N and let ∈ F R . Furthermore, let ∈ be a dire
tion and
u d F
u
let L be the set of lines in dire
tion in R . Then, the (dis
rete parallel) X-ray of in
u X F : := 0
u u 0
dire
tion is the fun
tion L −→ N N∪{ }, de(cid:28)ned by
X F(ℓ) := card(F ℓ) = (x).
u F
∩ 1
x ℓ
X∈
X F
u u
Moreover, the support of , i.e., the set of lines in L whi
h passthrough at least one point
F supp(X F) z d ℓz
of , is denoted by u . For ∈R , we denote by u the element of Lu whi
h passes
z
through .
supp(X F)
u
Remark 1. InthesituationofDe(cid:28)nition1, is(cid:28)niteand,moreover, the
ardinality
F X
of is impli
it in the -ray, sin
e one has
X F(ℓ)= card(F).
u
ℓ∈supXp(XuF)
d u Sd 1 F,F ( d)
− ′
Lemma 1. Let ∈N and let ∈ be a dire
tion. If ∈ F R , one has:
X F =X F card(F) = card(F )
u u ′ ′
(a) implies .
X F = X F F F u
u u ′ ′
(b) If , the
entroids of and lie on the same line parallel to .
(cid:3)
Proof. See [17, Lemma 5.1 and Lemma 5.4℄.
d 2 ( d) m U
De(cid:28)nition 2. Let ≥ , let E ⊂ F R , and let ∈ N. Further, let be a (cid:28)nite set of
1 T d
dire
tions and let T be a (cid:28)nite set of -dimensional subspa
es of R .
X U F,F
′
(a) We say that E is determined by the -rays in the dire
tions of if, for all ∈ E,
one has
(X F = X F u U) = F = F .
u u ′ ′
∀ ∈ ⇒
(b) We say that E is determined by the (orthogonal) proje
tions on the othogonal
om-
T T F,F
⊥ ′
plements of the subspa
es of T if, for all ∈ E, one has
(F T = F T T ) = F = F .
⊥ ′ ⊥ ′
| | ∀ ∈ T ⇒
X U
(
) We say that E is su

essively determined by the -rays in the dire
tions of , say
U = u ,...,u F
1 m
{ }, if, for a given ∈ E, these
an be
hosen indu
tively (i.e., the
u X F k 1,...,j 1 F

hoi
e of j depending on all uk with ∈ { − }) su
h that, for all ′ ∈ E,
one has
(X F = X F u U) = F = F .
u ′ u ′
∀ ∈ ⇒
DISCRETE TOMOGRAPHY 7
(d) We say that E is su

essively determined by the (orthogonal) proje
tions on the othog-
T T = T ,...,T
⊥ 1 m
onal
omplements of the subspa
es of T , say T { }, if, for a given
F T F T
∈ E, these
an be
hosen indu
tively (i.e., the
hoi
e of j depending on all | k⊥
k 1,...,j 1 F
′
with ∈ { − }) su
h that, for all ∈ E, one has
(F T = F T T ) = F = F .
⊥ ′ ⊥ ′
| | ∀ ∈ T ⇒
m X
(e) We say that E is determined (resp., su

essively determined) by -rays if there
U m
is a set of pairwise non-parallel dire
tions su
h that E is determined (resp.,
X U
su

essively determined) by the -rays in the dire
tions of .
m
(f) We say that E is determined (resp., su

essively determined) by (orthogonal) pro-
1 m
je
tions on orthogonal
omplements of -dimensional subspa
es if there isaset T of
1 d
pairwise non-parallel -dimensional subspa
es of R su
h that E is determined (resp.,
su

essively determined) by the (orthogonal) proje
tions on the othogonal
omple-
T T
⊥
ments of the subspa
es of T.
( d) X
Remark 2. Let E ⊂ F R . Note that if E is determined by a set of -rays (resp., proje
-
X
tions), then E is su

essively determined by the same -rays (resp., proje
tions).
2
De(cid:28)nition 3. Consider the Eu
lidean plane, R .
Ψ : 2 2
(a) A linear transformation (resp., a(cid:30)ne transformation) R −→ R is given by
z Az z Az +t A 2 2 t 2
7−→ (resp., 7−→ ), where is a real × matrix and ∈ R . In both
Ψ det(A) = 0 det(A) = 0

ases, is
alled singular (resp., non-singular) when (resp., 6 ).
h: 2 2 z λz +t λ
(b) A homothety R −→ R is given by 7−→ , where ∈ R is positive and
t 2 t = 0
∈ R . A homothety is
alled a dilatation if . Further, we
all a homothety
λ> 1
expansive if .
S 2
De(cid:28)nition 4. Let ⊂ R .
2
(a) A
onvex polygon is the
onvex hull of a (cid:28)nite set of points in R .
S S
(b) A polygon in is a
onvex polygon with all verti
es in .
C S S
(
) A (cid:28)nite subset of is
alled a
onvex set in if its
onvex hull
ontains no new
S C = conv(C) S S
points of , i.e., if ∩ . Moreover, the set of all
onvex sets in is
(S)
denoted by C .
(d) A regular polygon is always assumed to be planar, non-degenerate and
onvex. An
a(cid:30)nely regular polygon is a non-singular a(cid:30)ne image of a regular polygon. In parti
-
3
ular,Uit mSu1st have at least verti
es. P
(e) Let ⊂ be a (cid:28)nite set of dire
tions. We
all a non-degenerate
onvex polygon a
U v P u U
-polygon if it has the property that whenever is a vertex of and ∈ , the line
ℓv u v v
u ′
in the plane in dire
tion whi
h passes through also meets another vertex of
P
.
U
Remark 3. Note that -polygons have an even number of verti
es. Moreover, an a(cid:30)nely
U
regular polygon with an even number of verti
es is a -polygon if and only if ea
h dire
tion
U
of is parallel to one of its edges.
The following property is straight-forward.
h: 2 2 U S1
Lemma 2. Let R −→ R be a homothety and let ⊂ be a (cid:28)nite set of dire
tions.
Then, one has:
8 C.HUCK
P U h[P] P h U
(a) If is a -polygon, the image of under is again a -polygon.
F F ( 2) X U
1 2
(b) If ha[Fnd] arhe[Fele]ments of F R withXthe same -rays in the diUre
tions of , th(cid:3)e
1 2
sets and also have the same -rays in the dire
tions of .
n ζ n ζ =e2πi/n (ζ )
n n n
Forall ∈ N,and a(cid:28)xed primitive throot of unit(yζ (e+.gζ¯.,) ), letQ bethe
n n

orresponding
y
lotomi
(cid:28)eld. It is well known that Q is the maximal real sub(cid:28)eld
(ζ )
n
of Q ; see [41℄. Throughout this paper, we shall use the notation
= (ζ ), = (ζ +ζ¯ ), = [ζ ], = [ζ +ζ¯ ],
n n n n n n n n n n
K Q k Q O Z O Z
φ
while denotes Euler's phi-fun
tion (often also
alled Euler's totient fun
tion), i.e.,
φ(n) = card k 1 k n (k,n) = 1 .
∈ N| ≤ ≤ and
(cid:0)(cid:8) 2 (cid:9)(cid:1)
O

asionally, we shall identify C with R . Moreover, we denote the set of primes by P and
p 2p+1
its subset of Sophie Germain prime numbers (i.e., primes for whi
h the number is
SG
prime as well) by P .
p p
SG
Remark 4. Sophie Germain prime numbers ∈ P are the primes su
h that the equation
φ(n) = 2p
has solutions. It is not known whether there are in(cid:28)nitely many Sophie Germain
primes. The (cid:28)rst few are
2,3,5,11,23,29,41,53,83,89,113,131,173,
{
179,191,233,239,251,281,293,359,419,... ,
}
see entry A005384 of [39℄ for further details.
n 3
Lemma 3. For ≥ , one has:
2 = + ζ 1,ζ
n n n n n n n
(a) O is an O -module of rank . More pre
isely, O O O , and { } is an
n n
O -basis of O .
2 = + ζ 1,ζ
n n n n n n n
(b) K is a k -ve
tor spa
e of dimension . More pre
isely, K k k , and { }
n n
is a k -basis of K .
1,ζ
n n
Proof. First, we show (a). The linear independen
e of { } over O is
lear (sin
e by our
n 3 1,ζ
n
assumption ≥ , { } is even linearly independent over R). It su(cid:30)
es to prove that all
ζm ζm = α+βζ α,β
non-negative integral powζ2er=s αn+saβtζisfy n α,nβfor suitable ∈On. Usiζ¯ng=indζu
1tion,
we are done iζf2w=e sh1ow+(ζn +ζ 1)ζ n for suitable ∈On. To this end, note n n− an(cid:3)d
observe that n − n n− n. Claim (b) follows from the same argument.
2 N
n
Remark 5. Seen as a point set of R , O has -fold
y
li
symmetry, where
N = N(n):= [n,2].
(1)
n 1,2 = =
1 2
Ex
eptfortheone-dimensional
ase( ∈ { }, whereO O Z)andthe
rystallographi
n 3,6 = n = 4
3 6 4

ases ( ∈ { } (triangular latti
e O O ) and (square latti
e O ), see Figure 1),
2
n
O is dense in R ; see [4, Remark 1℄.
Proposition 1 (Gauÿ). [Kn :Q] = φ(n), hen
e the set {1,ζn,ζn2,...,ζnφ(n)−1} is a Q-basis of
/ G( / ) =
n n n ∼
K . The (cid:28)eld extension K Q is a Galois extension with Abelian Galois group K Q
( /n ) a( n) ζ ζa
Z Z ×, where mod
orresponds to the automorphism given by n 7−→ n.
(cid:3)
Proof. See [41, Theorem 2.5℄.
DISCRETE TOMOGRAPHY 9
( /n ) = a( n) (a,n) = 1
×
Remark 6. Note the identity Z Z { mod | } and see [3, Table 3℄ for
G( / )
n
examples of the expli
it stru
ture of K Q . This referen
e also
ontains further material
n
and referen
es on the role of O in the
ontext of model sets.
E/F/K
Lemma 4 (Degree formula). Let be an extension of (cid:28)elds. Then
[E : K]= [E :F][F :K].
(cid:3)
Proof. See [28, Ch. V.1, Proposition 1.2℄.
n 3 [ : ] = φ(n)/2
n n
Cor1o,l(laζry+1ζ¯.)I,f(ζ +≥ζ¯,)2o,n.e..h,a(sζ k+ζ¯ )Qφ(n)/2 1 . It follows/that a Q-basis of k is given
n n n n n n − n
by { }. Moreover, k Q is a Galois extension with
G( / ) = ( /n ) / 1( n)
Abelian Galois group kn Q ∼ Z Z × {± mod }.
Proof. This is an immediate
onsequen
e of Lemma 3(b), Lemma 4, Proposition 1 and Galo(cid:3)is
theory.
Remark 7. The dimension statement of Corollary 1 also follows via Galois theory from
n n
Lemma 4 in
onne
tion with the fa
t that k is the (cid:28)xed (cid:28)eld of K with respe
t to the
id,¯. G( / ) ¯.
n
subgroup { } of K Q , where denotes
omplex
onjugation, i.e., the automorphism
ζ ζ 1
given by n 7−→ n− (re
all that kn is the maximal real sub(cid:28)eld of Kn).
m,n =
m n (m,n)
Lemma 5. If ∈ N, then K ∩K K .
(m,n)=
Proof. Theassertionfollowsfromsimilarargumentsasintheproofofthespe
ial
ase
1 (ζ ,ζ ) = =
m n m n
;
ompare [28, Ch. VI.3, Corollary 3.2℄. Here, one has to observe Q K K
[m,n]
K and then to employ the identity
φ(m)φ(n) = φ([m,n])φ((m,n))
(2)
φ (cid:3)
instead of merely using the multipli
ativity of the arithmeti
fun
tion .
m,n
Lemma 6. Let ∈ N. The following statements are equivalent:
m n
(i) K ⊂ K .
m n m 2 (mod4) m 2n
(ii) | , or ≡ and | .
m n
Proof. For dire
tion (ii) ⇒ (i), suppose(cid:28)rst | . This
learly implies the assertion. Se
ondly,
m 2 (mod4) m = 2o o m 2n
o n
if ≡ , say for a suitable odd number , and | , then K ⊂ K (due
on =
o 2o m
to | ). However, Proposition 1 shows that the in
lusion of (cid:28)elds K ⊂ K K
annot be
φ φ(m) = φ(2o) = φ(o)
proper sin
e we have, by means of the multipli
ativity of , . This gives
m n
K ⊂ K .
m n
For dire
tion (i) ⇒ (ii), suppose the in
lusion of (cid:28)elds K ⊂ K . Then, Lemma 5 implies
=
m (m,n)
K K , hen
e
φ(m) = φ((m,n));
(3)
φ φ(pj) = pj 1(p 1)
−
see Proposition 1 again. Using the multipli
ativity of together with −
p j (m,n) < m
for ∈ P and ∈ N, we see that, given the
ase , (3)
an only be ful(cid:28)lled if
m 2 (mod4) m 2n (m,n) = m
m n≡ and | . The remaining
ase is equivalent to the relatio(cid:3)n
| .
The following
onsequen
e is immediate.
10 C.HUCK
m,n
Corollary 2. Let ∈ N. The following statements are equivalent:
=
m n
(i) mK= nK .m n = 2m n m = 2n (cid:3)
(ii) , or is odd and , or is odd and .
m,n 2 (mod4) =
m n
Remark 8. Corollary 2 implies that, for 6≡ , one has the identity K K
m = n
if and only if .
m,n m,n 3
Lemma 7. Let ∈ N with ≥ . Then, one has:
= = m,n 3,4,6 .
m n m n
(a) k k ⇐⇒ K K or ∈ { }
m 3,4,6 .
m n m n
(b) k ⊂ k ⇐⇒ K ⊂ K or ∈ { }
= =:
m n
Proof. For
laim (a), let us suppose k k k (cid:28)rst. Then, Proposition 1 and Corollary 1
[ : ] = [ : ] = 2 =
m n m n (m,n)
imply that K k K k . Note that K ∩ K K is a
y
lotomi
(cid:28)eld
= = =
m n (m,n) m n

ontaining k. It follows from Lemma 4 that either K ∩ K K K K or
= = = = =
m n (m,n) m n
K ∩ K K k and hen
e k k k Q, sin
e the latter is the only real
m,n 3,4,6

y
lotomi
(cid:28)eld. Now, this implies ∈ { }; see also Lemma 8(a) below. The othe(cid:3)r
dire
tion is obvious. Claim (b) follows immediately from the part (a).
φ n n 2 (mod4)
Lemma 8. Consider on { ∈ N| 6≡ }. Then, one has:
φ(n)/2 = 1 n 3,4
(a) if and only if ∈ { }. These are the
rystallographi

ases of the plane.
φ(n)/2 n := 8,9,12 2p+1 p
SG
(b) ∈ P if and only if ∈ S { }∪{ | ∈ P }.
φ
Pφ(rpojo)f.=Thpej e1q(upival1en)
es fpollow fromjthe multipli
ativity of in
onjun
tion with the identit(cid:3)y
−
− for ∈ P and ∈ N.
n 2 (mod4) /
n
Remark 9. Let 6≡ . By Corollary 1, k Q is a Galois extension with Abelian
G( / ) φ(n)/2 G( / )
n n
Galoisgroup k Q oforder . Now, itfollowsfromLemma 8that k Q istrivial
n 1,3,4 n
if and only if ∈ { }, and simple if and only if ∈ S.
n
Proposition 2. For ∈ N, one has:
n n
(a) O is the ring of
y
lotomi
integers in K , and hen
e its maximal order.
n n
(b) O is the ring of integers of k , and hen
e its maximal order.
(cid:3)
Proof. See [41, Theorem 2.6 and Proposition 2.16℄.
n
Remark 10. It follows from Proposition 2(a) and Proposition 1 that O is a Z-module of
rank φ(n) with Z-basis {1,ζn,ζn2,...,φζ(nφn(n))/−21}. Likewise, P1r,o(pζos+itioζ¯n),2((ζb)+anζ¯d)2C,o.r.o.l,l(aζry+1
n n n n n n
iζ¯m)pφl(yn)t/h2at1 O is a Z-module of rank with Z-basis {
n −
}.
λ
De(cid:28)nition 5. Let be a real algebrai
integer.
λ λ> 1
(a) We
all aPisot-Vijayaraghavannumber(PV-number)if whileallits(algebrai
)
1

onjugates have moduli stri
tly less than .
λ λ
(b) We
all a Pisot-Vijayaraghavan unit (PV-unit) if is a PV-number and is also a
1/λ
unit, i.e., if is an algebrai
integer as well.
n 1,2 ( ) φ(n)/2
n
Lemma 9. If ∈ N\{ }, then there is a PV-number of full degree in O .
(cid:3)
Proof. This follows immediately from [34, Ch. I, Theorem 2℄.