Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order

UNIQUENESS IN DISCRETE TOMOGRAPHY OF DELONE SETS WITH LONG-RANGE ORDER CHRISTIAN HUCK Abstra t. W e address the problem of determining nite subsets of Delone sets Λ ⊂ R d with long-range order b y X -ra ys in presrib ed Λ -diretions, i.e., diretions parallel to non-zero in terp oin t v etors of Λ . Here, an X -ra y in diretion u of a nite set giv es the n um b er of p oin ts in the set on ea h line parallel to u . F or our main result, w e in tro due the notion of algebrai Delone sets Λ ⊂ R 2 and deriv e a suien t ondition for the determination of the on v ex subsets of these sets b y X -ra ys in four presrib ed Λ -diretions. 1. Intr odution Disr ete tomo gr aphy is onerned with the in v erse problem of retrieving infor- mation ab out some nite ob jet from (generally noisy) information ab out its slies. An imp ortan t problem is the r e  onstrution of a nite p oin t set from its line sum funtions in a small n um b er of diretions. More preisely , a ( disr ete p ar al lel ) X- r ay of a nite subset of R d in diretion u giv es the n um b er of p oin ts in the set on ea h line parallel to u . In the traditional setting, motiv ated b y rystals, the p osi- tions to b e determined form a subset of a ommon translate of a lattie in R d . In fat, man y of the problems in disrete tomograph y ha v e b een studied on the square lattie; see [11 ℄, [12 ℄, [13 ℄, [16 ℄ and [19 ℄. In the longer run, b y also ha ving other strutures than p erfet rystals in mind, one has to tak e in to aoun t wider lasses of sets, or at least signian t deviations from the lattie struture. As an in terme- diate step b et w een p erio di and random (or amorphous) Delone sets , w e onsider Delone sets with long-r ange or der , th us inluding systems of ap erio di or der lik e mo del sets (also alled mathemati al quasirystals ) as a sp eial ase. The main motiv ation for our in terest in the disrete tomograph y of Delone sets Λ with long-range order omes from the fat that these sets serv e as a rather general mo del of atomi p ositions in solid state materials together with the demand of materials siene to reonstrut su h three-dimensional strutures or planar la y ers of them from their images under quan titativ e high r esolution tr ansmission ele tr on mir os opy (HR TEM). In fat, in [ 28 ℄ and [ 34 ℄ a te hnique is desrib ed, whi h, for some rystals, an eetiv ely measure the n um b er of atoms lying on densely o upied lines. Therefore, only Λ -diretions, i.e., diretions parallel to non-zero in terp oin t v etors of Λ , will b e onsidered. F urther, sine t ypial ob jets ma y b e damaged or ev en destro y ed b y the radiation energy after ab out 3 to 5 images tak en b y HR TEM, one is restrited to a small n um b er of X -ra ys. It atually is this restrition to few high-density dir e tions that mak es the problems of disrete tomograph y mathematially  hallenging, ev en if one assumes the absene of noise. Sine the ab o v e reonstrution problem of disrete tomograph y an p ossess rather dieren t solutions, one is led to the determination of nite subsets of a xed Delone set Λ b y X -ra ys in a small n um b er of suitably presrib ed Λ -diretions. The author w as supp orted b y the German Resear h Counil (Deuts he F ors h ungsgemein- s haft), within the CR C 701, and b y EPSR C via Gran t EP/D058465/1. 1 2 CHRISTIAN HUCK More preisely , w e sa y that the elemen ts of a olletion E of nite sets are deter- mine d b y the X -ra ys in a nite set of diretions if dieren t sets F and F ′ in E annot ha v e the same X -ra ys in these diretions. After summarizing a few general results on determination, w e in tro due the lass of algebr ai Delone sets Λ ⊂ R 2 (see Denition 4.1 ) and study the determination of the  onvex subsets of these sets b y X -ra ys in Λ -diretions. They are nite subsets of Λ with the prop ert y that their on v ex h ull on tains no new p oin ts of Λ . By using p -adi v aluation metho ds as in tro dued in disrete tomograph y b y Gardner and Gritzmann in [11 ℄ together with standard fats from the theory of (ylotomi) elds, w e deriv e a suien t ondition for the determination of the on v ex subsets of Λ b y X -ra ys in four Λ -diretions and sho w that three Λ -diretions nev er sue for this purp ose; f. Theorem 4.21 . F urther, b y using standard fats from algebrai n um b er theory and the theory of Pisot-Vija y aragha v an n um b ers, it is sho wn that ylotomi mo del sets (see Denition 4.22 ) are examples of algebrai Delone sets; f. Prop osition 4.31 . W e onlude with a disussion of the result on the determina- tion of on v ex sets b y four X -ra ys for this sp ei lass of ob jets; f. Theorem 4.33 . 2. Preliminaries and Not a tion Natural n um b ers are alw a ys assumed to b e p ositiv e and w e denote b y P the set of rational primes. W e denote the norm in Eulidean d -spae R d b y k · k . The unit sphere in R d is denoted b y S d − 1 and its elemen ts are also alled dir e tions . F or r > 0 and x ∈ R d , B r ( x ) is the op en ball of radius r ab out x . F or a subset S ⊂ R d , a diretion u ∈ S d − 1 is alled an S -dir e tion if it is parallel to a non-zero elemen t of the dierene set S − S of S . A homothety h : R d → R d is giv en b y z 7→ λz + t , where λ ∈ R is p ositiv e and t ∈ R d . A  onvex p olygon is the on v ex h ull of a nite set of p oin ts in R 2 . F or a subset S ⊂ R 2 , a p olygon in S is a on v ex p olygon with all v erties in S . F urther, a b ounded subset C of S is alled a  onvex subset of S if its on v ex h ull on tains no new p oin ts of S , i.e., if C = conv( C ) ∩ S . Let U ⊂ S 1 b e a nite set of diretions. A non-degenerate on v ex p olygon P is alled a U -p olygon if it has the prop ert y that whenev er v is a v ertex of P and u ∈ U , the line in the plane in diretion u whi h passes through v also meets another v ertex v ′ of P . Oasionally , w e iden tify C with R 2 and ¯ z will alw a ys denote the omplex onjugate of z . Consider a set Λ ⊂ R d , where d ∈ N . Λ is alled uniformly disr ete if there is a radius r > 0 su h that ev ery ball B r ( x ) with x ∈ R d on tains at most one p oin t of Λ . Note that the b ounded subsets of a uniformly disrete set Λ are preisely the nite subsets of Λ . Λ is alled r elatively dense if there is a radius R > 0 su h that ev ery ball B R ( x ) with x ∈ R d on tains at least one p oin t of Λ . Λ is alled a Delone set if it is b oth uniformly disrete and relativ ely dense. Λ is said to b e of nite lo  al  omplexity if Λ − Λ is disrete and losed. Note that a subset Λ has nite lo al omplexit y if and only if for ev ery r > 0 there are, up to translation, only nitely man y p athes of r adius r , i.e., sets of the form Λ ∩ B r ( x ) , where x ∈ R d ; f. [29 ℄. A Delone set Λ is a Meyer set if Λ − Λ is uniformly disrete. T ranslates Λ of arbitrary latties L ⊂ R d , are simple examples of Mey er sets, sine Λ − Λ = L is a Delone set. T rivially , an y Mey er set is of nite lo al omplexit y . Finally , Λ is alled ap erio di if it has no non-zero translation symmetries. Denition 2.1. Let F b e a nite subset of R d . F urthermore, let u ∈ S d − 1 b e a diretion and let L d u b e the set of lines in diretion u in R d . Then the ( disr ete p ar al lel ) X-r ay of F in dir e tion u is the funtion X u F : L d u → N 0 := N ∪ { 0 } , dened b y X u F ( ℓ ) := card( F ∩ ℓ ) = X x ∈ ℓ 1 F ( x ) . UNIQUENESS IN DISCRETE TOMOGRAPHY OF DELONE SETS 3 Moreo v er, the supp ort ( X u F ) − 1 ( N ) of X u F , i.e., the set of lines in L d u whi h pass through at least one p oin t of F , is denoted b y supp( X u F ) . F urther, for a nite set U ⊂ S d − 1 of diretions, set G F U := \ u ∈ U  [ ℓ ∈ supp( X u F ) ℓ  . Note that, in the situation of Denition 2.1 , one has F ⊂ G F U . F urther, G F U is nite if card U ≥ 2 . F at 2.2. L et h : R d → R d b e a homothety, and let U ⊂ S d − 1 b e a nite set of dir e tions. Then, one has: (a) If P is a U -p olygon, then h ( P ) is again a U -p olygon. (b) If F and F ′ ar e nite subsets of R d with the same X -r ays in the dir e tions of U , then the nite sets h ( F ) and h ( F ′ ) also have the same X -r ays in the dir e tions of U . Denition 2.3. Let E b e a olletion of nite subsets of R d and let U ⊂ S d − 1 b e a nite set of diretions. W e sa y that the elemen ts of E are determine d b y the X -ra ys in the diretions of U if, for all F, F ′ ∈ E , one has ( X u F = X u F ′ ∀ u ∈ U ) = ⇒ F = F ′ . 3. General resul ts on determina tion W e need the follo wing prop ert y of sets Λ ⊂ R d , where h S i Z denotes the Z -linear h ull of a set S ⊂ R d . In other w ords, h S i Z is the Ab elian group generated b y S . (Hom ∗ ) F or all nite subsets F of h Λ − Λ i Z , there is a homothet y h : R d → R d su h that h ( F ) ⊂ Λ . T ranslates Λ of arbitrary latties L ⊂ R d satisfy h Λ − Λ i Z = L and are th us De- lone sets with prop ert y (Hom ∗ ). The follo wing negativ e results sho ws that, in order to obtain p ositiv e results on determination for Delone sets Λ with prop ert y (Hom ∗ ), one has to imp ose some restrition on the nite subsets of Λ to b e determined. Prop osition 3.1. L et Λ ⊂ R d b e a Delone set with pr op erty (Hom ∗ ) and let U b e a nite set of p airwise non-p ar al lel Λ -dir e tions. Then the nite subsets of Λ ar e not determine d by the X -r ays in the dir e tions of U . Pr o of. Sine prop ert y (Hom ∗ ) is in v arian t under translations, w e ma y assume, with- out loss of generalit y , that 0 ∈ Λ . Hene, h Λ i Z ⊂ h Λ − Λ i Z . W e argue b y indution on card U . The ase card U = 0 means U = ∅ and is ob vious. Fix k ∈ N 0 and supp ose the assertion is true whenev er card U = k . Let U no w b e a set with card U = k + 1 . By the indution h yp othesis, there are dieren t nite subsets F and F ′ of Λ with the same X -ra ys in the diretions of U ′ , where U ′ ⊂ U satises card U ′ = k . Let u b e the remaining diretion of U and  ho ose a non-zero elemen t z ∈ h Λ − Λ i Z parallel to u su h that z + ( F ∪ F ′ ) and F ∪ F ′ are disjoin t. Then, F ′′ := F ∪ ( z + F ′ ) and F ′′′ := F ′ ∪ ( z + F ) are dieren t nite subsets of h Λ − Λ i Z with the same X -ra ys in the diretions of U . By prop ert y (Hom ∗ ), there is a ho- mothet y h : R d → R d su h that h ( F ′′ ∪ F ′′′ ) = h ( F ′′ ) ∪ h ( F ′′′ ) ⊂ Λ . It follo ws from F at 2.2 (b) that h ( F ′′ ) and h ( F ′′′ ) are dieren t nite subsets of Λ with the same X -ra ys in the diretions of U .  Remark 3.2. F or other v ersions of the last result, ompare [12 , Theorem 4.3.1℄ and [10 , Lemma 2.3.2℄. An analysis of the pro of of Prop osition 3.1 sho ws that, for an y Delone set Λ ⊂ R d with prop ert y (Hom ∗ ) and for an y nite set U of k pairwise non-parallel Λ -diretions, there are disjoin t nite subsets F and F ′ of Λ 4 CHRISTIAN HUCK Figure 1. T w o on tiguous subsets of Λ AB with the same X -ra ys in the t w o Λ AB -diretions with slop es 0 and 1 , resp etiv ely . with card F = car d F ′ = 2 ( k − 1) that ha v e the same X -ra ys in the diretions of U . Consider an y on v ex subset C of R d whi h on tains F and F ′ from ab o v e. Then the nite subsets F 1 := ( C ∩ Λ ) \ F and F 2 := ( C ∩ Λ ) \ F ′ of Λ also ha v e the same X -ra ys in the diretions of U . Whereas the p oin ts in F and F ′ are widely disp ersed o v er a region, those in F 1 and F 2 are on tiguous in a w a y similar to atoms in some solid state material. This pro edure is illustrated in Figure 1 in the ase of the ap erio di ylotomi mo del set Λ AB as desrib ed in Example 4.24 b elo w. Note that ylotomi mo del sets ha v e prop ert y (Hom ∗ ); see Denition 4.1 , Remark 4.2 and Prop osition 4.31 b elo w. The subsequen t p ositiv e results on determination are of limited use in pratie sine, in general, they do not omply with the restrition to few high-densit y di- retions men tioned earlier. The rst one is w ell-kno wn and follo ws from the same argumen ts as in the pro of of [12 , Theorem 4.3.3℄. F at 3.3. L et d ≥ 2 and let Λ b e a Delone set in R d . F urther, let U b e any set of k + 1 p airwise non-p ar al lel Λ -dir e tions wher e k ∈ N 0 . Then the nite subsets of Λ with  ar dinality less than or e qual to k ar e determine d by the X -r ays in the dir e tions of U . Mor e over, al l nite subsets F of Λ with  ar dinality less than or e qual to k satisfy F = G F U . Let d ≥ 2 and let Λ b e a Delone set in R d with prop ert y (Hom ∗ ). Remark 3.2 and F at 3.3 sho w that the nite subsets of Λ with ardinalit y less than or equal to k are determined b y the X -ra ys in an y set of k + 1 pairwise non-parallel Λ -diretions but not b y the X -ra ys in 1 + ⌊ log 2 k ⌋ pairwise non-parallel Λ -diretions. F at 3.4. L et d ≥ 2 and let Λ ⊂ R d b e r elatively dense. Then the set of Λ -dir e tions is dense in S d − 1 . Prop osition 3.5. L et d ≥ 2 , let r > 0 , and let Λ ⊂ R d b e a Delone set of nite lo  al  omplexity. Then, ther e is a set U of two non-p ar al lel Λ -dir e tions suh that the subsets of p athes of r adius r of Λ ar e determine d by the X -r ays in the dir e tions of U . Mor e over, ther e is a set U of thr e e p airwise non-p ar al lel Λ -dir e tions suh that, for al l subsets F of p athes of r adius r of Λ , one has F = G F U . UNIQUENESS IN DISCRETE TOMOGRAPHY OF DELONE SETS 5 Pr o of. W e denote b y P r ( Λ ) the olletion of subsets of pat hes of radius r of Λ . F or the rst assertion, note that the nite lo al omplexit y of Λ implies that the set V of Λ -diretions v with the prop ert y that there is a set F ∈ P r ( Λ ) and a line ℓ in R d in diretion v with more than one p oin t of F on ℓ is nite. Let u b e an arbitrary Λ -diretion. Then, for ev ery F ∈ P r ( Λ ) , one has F ⊂ G F { u } ∩ Λ . Cho ose u ′′ ∈ S d − 1 ∩ u ⊥ and note that, for ev ery F ∈ P r ( Λ ) , the orthogonal pro jetion ( G F { u } ∩ Λ ) | u ⊥ of the set G F { u } ∩ Λ on the h yp erplane u ⊥ is nite with diameter D F u < 2 r . Moreo v er, b y the nite lo al omplexit y of Λ , the set { D F u | F ∈ P r ( Λ ) } of diameters is nite. This implies the existene of a neigh b ourho o d W of u ′′ in S d − 1 with the prop ert y that, for ea h line ℓ in a diretion w ∈ W and an y set F ∈ P r ( Λ ) , an y t w o elemen ts of the set ℓ ∩ ( G F { u } ∩ Λ ) ha v e a distane less than 2 r . Sine the set of Λ -diretions is dense in S d − 1 b y F at 3.4 , and b y the niteness of the set V , this observ ation sho ws that one an  ho ose a Λ -diretion u ′ ∈ W \ V that is not parallel to u . W e laim that the elemen ts of P r ( Λ ) are determined b y the X -ra ys in the diretions of { u, u ′ } . T o this end, let F, F ′ ∈ P r ( Λ ) satisfy X u F = X u F ′ . Then, one has F, F ′ ⊂ G F { u } ∩ Λ . In order to demonstrate that the iden tit y X u ′ F = X u ′ F ′ implies the equalit y F = F ′ , it sues to sho w that ea h line ℓ in diretion u ′ meets at most one elemen t of G F { u } ∩ Λ . Assume the existene of t w o distint elemen ts, sa y λ and λ ′ , in ℓ ∩ ( G F { u } ∩ Λ ) . Then, b y onstrution, the distane b et w een λ and λ ′ is less than 2 r . Hene, { λ, λ ′ } ∈ P r ( Λ ) , and further u ′ ∈ V , a on tradition. F or the seond part, let u, u ′ b e t w o arbitrary non-parallel Λ -diretions and set U ′ := { u, u ′ } . Note that the nite lo al omplexit y of Λ implies that the set V of Λ -diretions v with the prop ert y that there is a set F ∈ P r ( Λ ) and a line ℓ in R d in diretion v with more than one p oin t of the nite set G F U ′ on ℓ is nite. Sine the set of Λ -diretions is dense in S d − 1 b y F at 3.4, this observ ation sho ws that one an  ho ose a Λ -diretion u ′′ / ∈ V that is not parallel to u and u ′ . By onstrution, the assertion follo ws with U := { u, u ′ , u ′′ } .  4. Determina tion of onvex subsets of algebrai Delone sets 4.1. Algebrai Delone sets. F or Λ ⊂ C , w e denote b y K Λ the eld extension of Q that is giv en b y K Λ := Q  Λ − Λ  ∪  Λ − Λ  , and, further, set k Λ := K Λ ∩ R , the maximal real subeld of K Λ . The follo wing notion will b e useful; see also [ 23, 24 ℄ for generalizations and results related to those presen ted b elo w. Denition 4.1. A Delone set Λ ⊂ R 2 is alled an algebr ai Delone set if it satises the follo wing prop erties: (Alg) [ K Λ : Q ] < ∞ . (Hom) F or all nite subsets F of K Λ , there is a homothet y h : R 2 → R 2 su h that h ( F ) ⊂ Λ . T ranslates Λ of the square lattie Z 2 = Z [ i ] are examples of algebrai Delone sets, with K Λ = Q ( i ) . Remark 4.2. Note that, for an y algebrai Delone set Λ , the eld k Λ is a real algebrai n um b er eld. T rivially , prop ert y (Hom) for Λ implies prop ert y (Hom ∗ ). Lagarias [26 ℄ dened the notion of nitely gener ate d Delone sets Λ ⊂ R d . These are Delone sets Λ with the prop ert y that the Ab elian group h Λ − Λ i Z is nitely gen- erated. The last prop ert y is alw a ys fullled b y Delone sets of nite lo al omplexit y , whi h are also alled Delone sets of nite typ e ; see [ 26 , Theorem 2.1℄. 6 CHRISTIAN HUCK Prop osition 4.3. L et Λ b e an algebr ai Delone set. If Λ − Λ is  ontaine d in the ring of algebr ai inte gers, then Λ is a nitely gener ate d Delone set. Pr o of. If Λ − Λ is on tained in the ring of algebrai in tegers, then the Ab elian group h Λ − Λ i Z is a subgroup of the ring of in tegers in K Λ . Sine the latter is the maximal order of K Λ and th us a free Ab elian group of nite rank, the assertion follo ws; f. [8, Ch. 2, Se. 2℄.  4.2. U -p olygons in algebrai Delone sets. The follo wing fat follo ws immedi- ately from prop ert y (Hom) in onjuntion with F at 2.2 (a). F at 4.4. L et Λ b e an algebr ai Delone set and let U ⊂ S 1 b e a nite set of dir e tions. Then ther e is a U -p olygon in K Λ if and only if ther e is a U -p olygon in Λ . Lemma 4.5. L et Λ b e an algebr ai Delone set. If U is any set of up to 3 p airwise non-p ar al lel Λ -dir e tions, then ther e exists a U -p olygon in Λ . Pr o of. Without loss of generalit y , w e ma y assume that card U = 3 . First, onstrut a triangle in K Λ ha ving sides parallel to the giv en diretions of U . If t w o of the v erties are  hosen in K Λ , then the third is automatially in K Λ . No w, t six ongruen t v ersions of this triangle together in the ob vious w a y to mak e an anely regular hexagon in K Λ . The latter is then a U -p olygon in K Λ and the assertion follo ws from F at 4.4 .  Prop osition 4.6. L et Λ b e an algebr ai Delone set and let U b e a set of two or mor e p airwise non-p ar al lel Λ -dir e tions. The fol lowing statements ar e e quivalent: (i) The  onvex subsets of Λ ar e determine d by the X -r ays in the dir e tions of U . (ii) Ther e is no U -p olygon in Λ . Pr o of. F or (i) ⇒ (ii), supp ose the existene of a U -p olygon P in Λ . P artition the v erties of P in to t w o disjoin t sets V , V ′ , where the elemen ts of these sets alternate round the b oundary bd P of P . Sine P is a U -p olygon, ea h line in the plane parallel to some u ∈ U that on tains a p oin t in V also on tains a p oin t in V ′ . In partiular, one sees that card V = card V ′ . Set C := ( Λ ∩ P ) \ ( V ∪ V ′ ) . Then, F := C ∪ V and F ′ := C ∪ V ′ are dieren t on v ex subsets of Λ with the same X -ra ys in the diretions of U . F or (ii) ⇒ (i), supp ose the existene of t w o dieren t on v ex subsets of Λ with the same X -ra ys in the diretions of U . Sine the prop ert y of b eing an algebrai Delone set is in v arian t under translations, w e ma y assume, without loss of generalit y , that 0 ∈ Λ , whene Λ ⊂ Λ − Λ . Then, b y the same argumen tation as in the pro of of the orresp onding diretion of [11 , Theorem 5.5℄, there follo ws the existene of a U -p olygon in Q ( Λ ) ⊂ K Λ . More preisely , one has to use Lemma 4.5 instead of [11 , Lemma 4.4℄ and note that [11 , Lemma 5.2℄ extends to the more general situation needed here. F at 4.4 ompletes the pro of.  Let ( t 1 , t 2 , t 3 , t 4 ) b e an ordered tuple of four pairwise distint elemen ts of the set R ∪ {∞} . Then, its r oss r atio h t 1 , t 2 , t 3 , t 4 i is the non-zero real n um b er dened b y h t 1 , t 2 , t 3 , t 4 i := ( t 3 − t 1 )( t 4 − t 2 ) ( t 3 − t 2 )( t 4 − t 1 ) , where one uses the usual on v en tions if one of the t i equals ∞ . F at 4.7. F or a set Λ ⊂ R 2 , the r oss r atio of slop es of four p airwise non-p ar al lel Λ -dir e tions is an element of the eld k Λ . UNIQUENESS IN DISCRETE TOMOGRAPHY OF DELONE SETS 7 The pro of of the follo wing en tral result uses Darb oux's theorem on seond midp oin t p olygons; see [9 ℄, [14 ℄ or [10 , Ch. 1℄. Theorem 4.8. L et Λ ⊂ R 2 , let U b e a set of four or mor e p airwise non-p ar al lel Λ -dir e tions, and supp ose the existen e of a U -p olygon. Then the r oss r atio of slop es of any four dir e tions of U , arr ange d in or der of inr e asing angle with the p ositive r e al axis, is an element of the set (4.1)  [ m ≥ 4  [ k 3 1 while all (algebrai) onjugates of λ ha v e mo duli stritly less than 1 . The follo wing fat follo ws from [31 , Ch. 1, Theorem 2℄. F at 4.26. F or n ≥ 3 , ther e is a PV-numb er of de gr e e φ ( n ) / 2 in Z [ ζ n + ¯ ζ n ] . F or n ≥ 3 and λ ∈ Z [ ζ n + ¯ ζ n ] , w e denote b y m ⋆ λ the Z -mo dule endomorphism of Z [ ζ n ] ⋆ whi h is giv en b y m ⋆ λ ( z ⋆ ) = ( λz ) ⋆ , where z ∈ Z [ ζ n ] and . ⋆ is a star map of a ylotomi mo del set with underlying Z -mo dule Z [ ζ n ] . Lemma 4.27. L et n ∈ N \ { 1 , 2 , 3 , 4 , 6 } , and let . ⋆ b e a star map of a ylotomi mo del set with underlying Z -mo dule Z [ ζ n ] . Then, for any PV-numb er λ of de gr e e φ ( n ) / 2 in Z [ ζ n + ¯ ζ n ] , a suitable p ower of m ⋆ λ is  ontr ative, i.e., ther e is a k ∈ N and a ξ ∈ (0 , 1) suh that k ( m ⋆ λ ) k ( z ⋆ ) k ≤ ξ k z ⋆ k holds for al l z ∈ Z [ ζ n ] . 12 CHRISTIAN HUCK Figure 2. A en tral pat h of the eigh tfold Ammann-Beenk er tiling with v ertex set Λ AB (left) and the . ⋆ -image of Λ AB inside the o tagonal windo w (righ t), with relativ e sale as desrib ed in the text. Pr o of. Sine all norms on R d are equiv alen t, it sues to pro v e the assertion in ase of the maxim um norm k · k ∞ on ( R 2 ) φ ( n ) / 2 − 1 with resp et to the Eulidean norm on R 2 rather than onsidering the Eulidean norm k · k on ( R 2 ) φ ( n ) / 2 − 1 itself. But in that ase, the assertion follo ws immediately with k := 1 and ξ := max {| σ j ( λ ) | | j ∈ { 2 , . . . , φ ( n ) / 2 }} , sine the set { λ, σ 2 ( λ ) , . . . , σ φ ( n ) / 2 ( λ ) } equals the set of onjugates of λ .  Lemma 4.28. L et n ≥ 3 and let . ⋆ b e a star map of a ylotomi mo del set with underlying Z -mo dule Z [ ζ n ] . Then Z [ ζ n ] ⋆ is dense in ( R 2 ) φ ( n ) / 2 − 1 . Pr o of. If n ∈ { 3 , 4 , 6 } , one ev en has Z [ ζ n ] ⋆ = ( R 2 ) φ ( n ) / 2 − 1 = { 0 } . Otherwise, let λ b e a PV-n um b er of degree φ ( n ) / 2 in Z [ ζ n + ¯ ζ n ] ; f. F at 4.26 . Then, sine Z [ ζ n ] is the maximal order of Q ( ζ n ) b y Remark 4.23 , for an y k ∈ N , the set { ( λ k z , ( m ⋆ λ ) k ( z ⋆ )) | z ∈ Z [ ζ n ] } is a (full) lattie in R 2 × ( R 2 ) φ ( n ) / 2 − 1 ; f. [ 8 , Ch. 2, Se. 3℄. In onjuntion with Lemma 4.27 , this implies that, for an y ε > 0 , the Z -mo dule Z [ ζ n ] ⋆ on tains an R -basis of ( R 2 ) φ ( n ) / 2 − 1 whose elemen ts ha v e norms ≤ ε . The assertion follo ws.  Lemma 4.29. L et n ≥ 3 and let Λ b e a ylotomi mo del set with underlying Z - mo dule Z [ ζ n ] . Then, for any nite set F ⊂ Q ( ζ n ) , ther e is a homothety h : R 2 → R 2 suh that h ( F ) ⊂ Λ . Pr o of. Without loss of generalit y , w e ma y assume that Λ is of the form Λ ( W ) (see Denition 4.22 ) and, further, that F 6 = ∅ . By [27 , Ch. 7.1, Prop osition 1.1℄ in onjuntion with F at 4.12 and Remark 4.23 , there is an l ∈ N with l F ⊂ Z [ ζ n ] . Let . ⋆ b e the star map that is used in the onstrution of Λ ( W ) . If n ∈ { 3 , 4 , 6 } , w e are done b y dening the homothet y h : R 2 → R 2 b y z 7→ l z . Otherwise, b y int W 6 = ∅ in onjuntion with Lemma 4.28 , there follo ws the existene of a suitable z 0 ∈ Z [ ζ n ] with z ⋆ 0 ∈ in t W . Consider the op en neigh b ourho o d V := in t W − z ⋆ 0 of 0 in ( R 2 ) φ ( n ) / 2 − 1 . Next,  ho ose a PV-n um b er λ of degree φ ( n ) / 2 in Z [ ζ n + ¯ ζ n ] ; f. F at 4.26 . By virtue of Lemma 4.27 , there is a k ∈ N su h that ( m ⋆ λ ) k (( lF ) ⋆ ) ⊂ V . UNIQUENESS IN DISCRETE TOMOGRAPHY OF DELONE SETS 13 It follo ws that { ( λ k z + z 0 ) ⋆ | z ∈ l F } ⊂ int W and, further, that h ( F ) ⊂ Λ ( W ) , where h : R 2 → R 2 is the homothet y giv en b y z 7→ ( l λ k ) z + z 0 .  F at 4.30. L et n ≥ 3 and let Λ b e a ylotomi mo del set with underlying Z -mo dule Z [ ζ n ] . Then, one has K Λ ⊂ Q ( ζ n ) and thus k Λ ⊂ Q ( ζ n + ¯ ζ n ) . Prop osition 4.31. Cylotomi mo del sets ar e algebr ai Delone sets. Pr o of. An y ylotomi mo del set is a Mey er set b y Remark 4.23 . Prop erties (Alg) and (Hom) follo w immediately from F at 4.30 in onjuntion with F at 4.12 and Lemma 4.29 , resp etiv ely .  As another immediate onsequene of Lemma 4.29 , one v eries the follo wing F at 4.32. L et n ≥ 3 and let Λ b e a ylotomi mo del set with underlying Z -mo dule Z [ ζ n ] . Then the set of Λ -dir e tions is pr e isely the set of Z [ ζ n ] -dir e tions. An analysis of the pro of of Theorem 4.21 giv es the follo wing Theorem 4.33. F or al l n ≥ 3 , ther e is a nite set N n ⊂ Q suh that, for al l ylotomi mo del sets Λ with underlying Z -mo dule Z [ ζ n ] and al l sets U of four p airwise non-p ar al lel Z [ ζ n ] -dir e tions, one has the fol lowing: If U has the pr op erty that the r oss r atio of slop es of the dir e tions of U , arr ange d in or der of inr e asing angle with the p ositive r e al axis, do es not map under the norm N Q ( ζ n + ¯ ζ n ) / Q to N n , then the  onvex subsets of Λ ar e determine d by the X -r ays in the dir e tions of U . The ab o v e analysis allo ws the onstrution of sp ei sets U of four pairwise non-parallel Z [ ζ n ] -diretions ha ving the prop ert y that, for all ylotomi mo del sets Λ with underlying Z -mo dule Z [ ζ n ] , the on v ex subsets of Λ are determined b y the orresp onding X -ra ys. F or example, the sets of Z [ ζ n ] -diretions parallel to the elemen ts of the follo wing sets ha v e this prop ert y: U n := { 1 , 1 + ζ n , 1 + 2 ζ n , 1 + 5 ζ n } , U ′ n := { 1 , 2 + ζ n , ζ n , − 1 + 2 ζ n } and U ′′ n := { 2 + ζ n , 3 + 2 ζ n , 1 + ζ n , 2 + 3 ζ n } ; f. [21 , Theorem 2.54℄ or [20 , Theorem 15℄ and ompare [11 , Theorem 5.7 and Remark 5.8℄. Often, one an ev en nd examples that yield dense lines in the orresp onding disrete strutures. F or example, for the pratially relev an t quasirystallographi ase of (ap erio di) ylotomi mo del sets Λ with underlying Z - mo dule Z [ ζ n ] , where n = 5 , 8 , 10 , 12 , this is a hiev ed b y the sets of Z [ ζ n ] -diretions parallel to the elemen ts of the follo wing sets, where τ denotes the golden ratio (i.e., τ = (1 + √ 5) / 2 ): U 8 := { 1 + ζ 8 , ( − 1 + √ 2) + √ 2 ζ 8 , ( − 1 − √ 2) + ζ 8 , − 2 + ( − 1 + √ 2) ζ 8 } , U 5 := U 10 := { (1 + τ ) + ζ 5 , ( τ − 1) + ζ 5 , − τ + ζ 5 , 2 τ − ζ 5 } and U 12 := { 1 , 2 + ζ 12 , ζ 12 , √ 3 − ζ 12 } , resp etiv ely; f. [21 , Theorem 2.56, Example 2.57 and Remark 2.58℄ or [20 , Theorem 16, Example 3 and Remark 40℄. Note that orders 5 , 8 , 10 and 12 o ur as standard yli symmetries of gen uine quasirystals; f. [35 ℄. Problem 4.34. In the rystallographi ase of the square lattie Z [ i ] , Gardner and Gritzmann w ere able to sho w that the on v ex subsets of Z [ i ] are determined b y the X -ra ys in the diretions of any set U of sev en pairwise non-parallel Z [ i ] -diretions; f. [11 , Theorem 5.7℄. It w ould b e in teresting to kno w if, for all ylotomi mo del sets Λ with underlying Z -mo dule Z [ ζ n ] , there exists a natural n um b er k ∈ N su h that the on v ex subsets of Λ are determined b y the X -ra ys in the diretions of an y set U of k pairwise non-parallel Z [ ζ n ] -diretions. Assuming an armativ e answ er, a w eak relation b et w een n and k w as demonstrated in [23 , Corollary 5.5℄. Final remark F or a summary of results for mo del sets asso iated with the famous Penr ose tiling of the plane, see [ 3℄. These so-alled Penr ose mo del sets an also b e seen to b e algebrai Delone sets. The algorithmi r e  onstrution pr oblem of disrete 14 CHRISTIAN HUCK tomograph y of ylotomi mo del sets has b een studied in [2℄. In [22 ℄, it is sho wn ho w the results for the planar ase obtained in [ 2 ℄ and the presen t text an b e lifted to the pratially relev an t ase of so-alled i osahe dr al mo del sets in R 3 . F or a ompleter o v erview of b oth uniqueness and omputational omplexit y results in the disrete tomograph y of Delone sets with long-range order, w e refer the reader to [21 ℄. This referene also on tains results on the in terativ e onept of su  essive determination of nite sets b y X -ra ys and further extensions of settings and results that are b ey ond our sop e here; ompare also [17 ℄, [20 ℄ and [22 ℄. A kno wledgements The author is indebted to Mi hael Baak e, Ri hard J. Gardner and P eter A. B. Pleasan ts for their o op eration and for useful hin ts on the man usript. V aluable disussions with Uw e Grimm, P eter Gritzmann and Barbara Langfeld are gratefully a kno wledged. Referenes [1℄ R. Ammann, B. Grün baum and G. C. Shephard, Ap erio di tiles, Disr ete Comput. Ge om. 8 (1992), 125. [2℄ M. Baak e, P . Gritzmann, C. Hu k, B. Langfeld and K. Lord, Disrete tomograph y of planar mo del sets, A ta Crystal lo gr. A 62 (2006), 419433; arXiv:math/0609393v1 [math.MG℄ [3℄ M. Baak e and C. Hu k, Disrete tomograph y of P enrose mo del sets, Philos. Mag. 87 (2007), 28392846; arXiv:math-ph/0610056v1 [4℄ M. Baak e and D. Joseph, Ideal and defetiv e v ertex ongurations in the planar o tagonal quasilattie, Phys. R ev. B 42 (1990), 80918102. [5℄ M. Baak e, P . Kramer, M. S hlottmann and D. Zeidler, The triangle pattern  a new quasip eri- o di tiling with v efold symmetry , Mo d. Phys. L ett. B 4 (1990), 249258. [6℄ M. Baak e, P . Kramer, M. S hlottmann and D. Zeidler, Planar patterns with v efold symmetry as setions of p erio di strutures in 4-spae, Int. J. Mo d. Phys. B 4 (1990), 22172268. [7℄ M. Baak e and R. V. Mo o dy (eds.), Dir e tions in Mathemati al Quasirystals , CRM Mono- graph Series, v ol. 13, AMS, Pro videne, RI, 2000. [8℄ Z. I. Borevi h and I. R. Shafarevi h, Numb er The ory , A ademi Press, New Y ork, 1966. [9℄ M. G. Darb oux, Sur un problème de géométrie élémen taire, Bul l. Si. Math. 2 (1878), 298 304. [10℄ R. J. Gardner, Ge ometri T omo gr aphy , 2nd ed., Cam bridge Univ ersit y Press, New Y ork, 2006. [11℄ R. J. Gardner and P . Gritzmann, Disrete tomograph y: determination of nite sets b y X-ra ys, T r ans. A mer. Math. So . 349 (1997), 22712295. [12℄ R. J. Gardner and P . Gritzmann, Uniqueness and omplexit y in disrete tomograph y , in: [19 ℄, pp. 85114. [13℄ R. J. Gardner, P . Gritzmann and D. Prangen b erg, On the omputational omplexit y of reonstruting lattie sets from their X-ra ys, Disr ete Math. , 202 (1999), 4571. [14℄ R. J. Gardner and P . MMullen, On Hammer's X-ra y problem, J. L ondon Math. So . (2) 21 (1980), 171175. [15℄ F. Q. Gouv êa, p-adi Numb ers , Springer, New Y ork, 1993. [16℄ P . Gritzmann, On the reonstrution of nite lattie sets from their X-ra ys, in: L e tur e Notes on Computer Sien e , eds. E. Ahrono vitz and C. Fiorio, Springer, London, 1997, pp. 1932. [17℄ P . Gritzmann and B. Langfeld, On the index of Siegel grids and its appliation to the tomog- raph y of quasirystals, Eur op e an J. Combin. , 29 (2008), 18941909. [18℄ F. Gähler, Mat hing rules for quasirystals: the omp osition-deomp osition metho d, J. Non- Cryst. Solids 153-154 (1993), 160164. [19℄ G. T. Herman and A. Kuba (eds.), Disr ete T omo gr aphy: F oundations, A lgorithms, and Appli ations , Birkhäuser, Boston, 1999. [20℄ C. Hu k, Uniqueness in disrete tomograph y of planar mo del sets, notes (2007); arXiv:math/0701141v2 [math.MG℄ [21℄ C. Hu k, Disr ete T omo gr aphy of Delone Sets with L ong-R ange Or der , Ph.D. thesis (Univ er- sität Bielefeld), Logos V erlag, Berlin, 2007. [22℄ C. Hu k, Disrete tomograph y of iosahedral mo del sets, submitted; [math.MG℄ UNIQUENESS IN DISCRETE TOMOGRAPHY OF DELONE SETS 15 [23℄ C. Hu k, A note on anely regular p olygons, Eur op e an J. Combin. , in press; [math.MG℄ [24℄ C. Hu k, On the existene of U -p olygons of lass c ≥ 4 in planar p oin t sets, in preparation. [25℄ N. K oblitz, p-adi Numb ers, p-adi A nalysis, and Zeta-F untions , 2nd ed., Springer, New Y ork, 1984. [26℄ J. C. Lagarias, Geometri mo dels for quasirystals I. Delone sets of nite t yp e, Disr ete Comput. Ge om. 21 (1999), no. 2, 161191. [27℄ S. Lang, A lgebr a , 3rd ed., A ddison-W esley , Reading, MA, 1993. [28℄ C. Kisielo wski, P . S h w ander, F. H. Baumann, M. Seibt, Y. Kim and A. Ourmazd, An approa h to quan titativ e high-resolution transmission eletron mirosop y of rystalline ma- terials, Ultr amir os opy 58 (1995), 131155. [29℄ R. V. Mo o dy , Mo del sets: a surv ey , in: F r om Quasirystals to Mor e Complex Systems , eds. F. Axel, F. Déno y er and J.-P . Gazeau, EDP Sienes, Les Ulis, and Springer, Berlin, 2000, pp. 145166; arXiv:math/0002020v1 [math.MG℄ [30℄ P . A. B. Pleasan ts, Designer quasirystals: ut-and-pro jet sets with pre-assigned prop erties, in: [7℄, pp. 95141. [31℄ R. Salem, A lgebr ai Numb ers and F ourier A nalysis , D. C. Heath and Compan y , Boston, 1963. [32℄ M. S hlottmann, Cut-and-pro jet sets in lo ally ompat Ab elian groups, in: Quasirystals and Disr ete Ge ometry , ed. J. P atera, Fields Institute Monographs, v ol. 10, AMS, Pro videne, RI, 1998, pp. 247264. [33℄ M. S hlottmann, Generalized mo del sets and dynamial systems, in: [7 ℄, pp. 143159. [34℄ P . S h w ander, C. Kisielo wski, M. Seibt, F. H. Baumann, Y. Kim and A. Ourmazd, Mapping pro jeted p oten tial, in terfaial roughness, and omp osition in general rystalline solids b y quan titativ e transmission eletron mirosop y , Phys. R ev. L ett. 71 (1993), 41504153. [35℄ W. Steurer, T w en t y y ears of struture resear h on quasirystals. P art I. P en tagonal, o tagonal, deagonal and do deagonal quasirystals, Z. Kristal lo gr. 219 (2004), 391446. [36℄ L. C. W ashington, Intr o dution to Cylotomi Fields , 2nd ed., Springer, New Y ork, 1997. Dep ar tment of Ma thema tis and St a tistis, The Open University, W al ton Hall, Mil ton Keynes, MK7 6AA, United Kingdom E-mail addr ess : .hukopen.a.uk