Solution of a uniqueness problem in the discrete tomography of algebraic Delone sets
aa r X i v : . [ m a t h . M G ] J a n SOLUTION OF A UNIQUENESS PROBLEM IN THE DISCRETETOMOGRAPHY OF ALGEBRAIC DELONE SETS
CHRISTIAN HUCK AND MICHAEL SPIEß
Abstract.
We consider algebraic Delone sets Λ in the Euclidean plane andaddress the problem of distinguishing convex subsets of Λ by X-rays in pre-scribed Λ -directions, i.e., directions parallel to nonzero interpoint vectors of Λ . Here, an X-ray in direction u of a finite set gives the number of points inthe set on each line parallel to u . It is shown that for any algebraic Deloneset Λ there are four prescribed Λ -directions such that any two convex sub-sets of Λ can be distinguished by the corresponding X-rays. We further provethe existence of a natural number c Λ such that any two convex subsets of Λ can be distinguished by their X-rays in any set of c Λ prescribed Λ -directions.In particular, this extends a well-known result of Gardner and Gritzmann onthe corresponding problem for planar lattices to nonperiodic cases that arerelevant in quasicrystallography. Introduction
Discrete tomography is concerned with the inverse problem of retrieving informa-tion about some finite object in Euclidean space from (generally noisy) informationabout its slices. One important problem is the unique reconstruction of a finitepoint set in Euclidean -space from its (discrete parallel) X-rays in a small numberof directions, where the X-ray of the finite set in a certain direction is the line sumfunction giving the number of points in the set on each line parallel to this direction.The interest in the discrete tomography of planar Delone sets Λ with long-rangeorder is motivated by the requirement in materials science for the unique recon-struction of solid state materials like quasicrystals slice by slice from their imagesunder quantitative high resolution transmission electron microscopy (HRTEM). Infact, in [28], [35] a technique is described, which can, for certain crystals, effectivelymeasure the number of atoms lying on densely occupied columns. It is reason-able to expect that future developments in technology will extend this situation toother solid state materials. The aforementioned density condition forces us to con-sider only Λ -directions, i.e., directions parallel to nonzero interpoint vectors of Λ .Further, since typical objects may be damaged or even destroyed by the radiationenergy after about to images taken by HRTEM, applicable results may only usea small number of X-rays. It actually is this restriction to few high-density direc-tions that makes the problems of discrete tomography mathematically challenging,even if one assumes the absence of noise.In the traditional setting, motivated by crystals , the positions to be determinedform a finite subset of a three-dimensional lattice, the latter allowing a slicing intoequally spaced congruent copies of a planar lattice. In the crystallographic setting,by the affine nature of the problem, it therefore suffices to study the discrete tomog-raphy of the square lattice; cf. [14], [15], [16], [18], [20], [21], [22] for an overview. Mathematics Subject Classification.
Primary 52C23; Secondary 11R04, 11R06, 11R18,12F10, 52A10, 52C05,82D99.
Key words and phrases.
Discrete tomography, X-ray, algebraic Delone set, model set, U -polygon, convex body. For the quasicrystallographic setting, the positions to be determined form a finitesubset of a nonperiodic Delone set with long-range order (more precisely, a math-ematical quasicrystal or model set [8], [29]) which on the other hand is containedin a free additive subgroup of R of finite rank r > . These model sets possess,as it is the case for lattices, a dimensional hierarchy, i.e., they allow a slicing intoplanar model sets. However, the slices are in general no longer pairwise congruentor equally spaced in -space; cf. [31]. Still, most of the model sets that describe realquasicrystallographic structures allow a slicing such that each slice is, when seenfrom a common perpendicular viewpoint, a (planar) n -cyclotomic model set , where n = 5 , n = 8 and n = 12 , respectively (Example 3.11); cf. [23, Sec. 1.2], [25], [27,Sec. 4.5] and [36] for details. These cyclotomic model sets thus take over the roleplayed by the planar lattices in the crystallographic case. In the present text, weshall focus on the larger class of algebraic Delone sets (Definition 3.1).Since different finite subsets of a Delone set Λ may have the same X-rays inseveral Λ -directions (in other words, the above problem of uniquely reconstructinga finite point set from its X-rays is an ill-posed problem in general), one is naturallyinterested in conditions to be imposed on the set of Λ -directions together withrestrictions on the possible finite subsets of Λ such that the latter phenomenoncannot occur. Here, we consider the convex subsets of Λ (i.e., bounded subsets of Λ with the property that their convex hull contains no new points of Λ ) and show thatfor any algebraic Delone set Λ there are four prescribed Λ -directions such that anytwo convex subsets of Λ can be distinguished by the corresponding X-rays, whereasless than four Λ -directions never suffice for this purpose (Theorem 5.10(a)). Wefurther prove the existence of a finite number c Λ such that any two convex subsetsof Λ can be distinguished by their X-rays in any set of c Λ prescribed Λ -directions(Theorem 5.10(b)). Moreover, we demonstrate that the least possible numbers c Λ in the case of the practically most relevant examples of n -cyclotomic model sets Λ with n = 5 , n = 8 and n = 12 are (in that very order) , and (Theorem 5.11(b)and Remark 5.12). This extends a well-known result of Gardner and Gritzmann(cf. [14, Thm. 5.7]) on the corresponding problem for planar lattices Λ ( c Λ = 7 )to cases that are relevant in quasicrystallography and particularly solves Problem4.34 of [27]. The above results and their continuous analogue (Theorem 6.2) followfrom deep insights into the existence of certain U -polygons in the plane (cf. Sec. 2).We believe that our main result on these polygons (Theorem 5.6) is of independentinterest from a purely geometrical point of view. For the algorithmic reconstructionproblem in the quasicrystallographic setting, we refer the reader to [3], [25].2. Preliminaries and notation
Natural numbers are always assumed to be positive. We denote the norm inEuclidean d -space by k · k . The Euclidean plane will occasionally be identifiedwith the complex numbers. For z ∈ C , ¯ z denotes the complex conjugate of z and | z | = √ z ¯ z its modulus. The unit circle in C is denoted by S and its elementsare also called directions . For z ∈ C ∗ , we denote by sl( z ) the slope of z , i.e., sl( z ) = − i ( z − ¯ z ) / ( z + ¯ z ) ∈ R ∪ {∞} . For r > and z ∈ C , B r ( z ) is the openball of radius r about z . Recall that an ( R -) linear endomorphism (resp., affineendomorphism ) of C is given by z az + b ¯ z (resp., z az + b ¯ z + t ), where a, b, t ∈ C . In both cases, it is an automorphism if and only if az + b ¯ z = 0 onlyholds for z = 0 . A homothety h : C → C is given by z λz + t , where λ ∈ R ispositive and t ∈ C . In the following, let Λ be a subset of C . A direction u ∈ S is called a Λ -direction if it is parallel to a nonzero element of the difference set Λ − Λ = { v − w | v, w ∈ Λ } of Λ . A convex polygon is the convex hull of a finite setof points in C . A polygon in Λ is a convex polygon with all vertices in Λ . Further, NIQUENESS IN DISCRETE TOMOGRAPHY OF ALGEBRAIC DELONE SETS 3 a bounded subset C of Λ is called a convex subset of Λ if C = conv( C ) ∩ Λ , where conv( C ) denotes the convex hull of C . Let U ⊂ S be a finite set of directions.A nondegenerate convex polygon P is called a U -polygon if it has the propertythat whenever v is a vertex of P and u ∈ U , the line in the complex plane indirection u which passes through v also meets another vertex v ′ of P . By a regularpolygon we shall always mean a nondegenerate convex regular polygon. An affinelyregular polygon is the image of a regular polygon under an affine automorphism ofthe complex plane. Λ is called uniformly discrete if there is a radius r > suchthat every ball B r ( z ) with z ∈ C contains at most one point of Λ . Note that thebounded subsets of a uniformly discrete set Λ are precisely the finite subsets of Λ . Λ is called relatively dense if there is a radius R > such that every ball B R ( z ) with z ∈ C contains at least one point of Λ . Λ is called a Delone set if it is bothuniformly discrete and relatively dense. Λ is said to be of finite local complexity if Λ − Λ is discrete and closed. Note that Λ is of finite local complexity if and only iffor every r > there are, up to translation, only finitely many patches of radius r ,i.e., sets of the form Λ ∩ B r ( z ) , where z ∈ C ; cf. [29]. A Delone set Λ is a Meyer set if Λ − Λ is uniformly discrete. Trivially, any Meyer set is of finite local complexity. Λ is called periodic if it has nonzero translation symmetries. Finally, we denote by K Λ the intermediate field of C / Q that is given by K Λ := Q (cid:0) ( Λ − Λ ) ∪ (cid:0) Λ − Λ (cid:1)(cid:1) . Recollections from the theory of cyclotomic fields.
Let K ⊂ C be afield and let µ be the group of roots of unity in C . We denote the maximal realsubfield K ∩ R of K by K + and set µ ( K ) := µ ∩ K . For n ∈ N , we always let ζ n := e πi/n , a primitive n th root of unity in C . Then, Q ( ζ n ) is the n th cyclotomicfield. Further, φ will always denote Euler’s totient function, i.e., φ ( n ) = card (cid:0)(cid:8) k ∈ N (cid:12)(cid:12) ≤ k ≤ n and gcd( k, n ) = 1 (cid:9)(cid:1) . Recall that φ is multiplicative with φ ( p r ) = p r − ( p − for p prime and r ∈ N . Fact 2.1 (Gauß) . [37, Thm. 2.5] [ Q ( ζ n ) : Q ] = φ ( n ) and the field extension Q ( ζ n ) / Q is a Galois extension with Abelian Galois group G ( Q ( ζ n ) / Q ) ≃ ( Z /n Z ) × ,with a (mod n ) corresponding to the automorphism given by ζ n ζ an . (cid:3) Note that the composition Q ( ζ n ) Q ( ζ m ) = Q ( ζ n , ζ m ) of cyclotomic fields is equalto the cyclotomic field Q ( ζ lcm( n,m ) ) . Further, the intersection Q ( ζ n ) ∩ Q ( ζ m ) ofcyclotomic fields is equal to the cyclotomic field Q ( ζ gcd( n,m ) ) . Note that Q ( ζ n ) + = Q ( ζ n + ¯ ζ n ) = Q ( ζ n + ζ − n ) . Clearly, if n divides m then Q ( ζ n ) is a subfield of Q ( ζ m ) .Since Q ( ζ n ) = Q ( ζ n ) for odd n by Fact 2.1, we may sometimes restrict ourselvesto n ∈ N with n .2.2. Cross ratios.
Let ( t , t , t , t ) be an ordered tuple of four pairwise distinctelements of R ∪ {∞} . Then, its cross ratio h t , t , t , t i is the nonzero real numberdefined by h t , t , t , t i := ( t − t )( t − t )( t − t )( t − t ) , with the usual conventions if one of the t i equals ∞ . We need the following invari-ance property of cross ratios of slopes. Fact 2.2. [23, Lemma 2.17]
Let z , z , z , z ∈ C ∗ be pairwise nonparallel and let Ψ be a linear automorphism of the complex plane. Then, one has (cid:10) sl( z ) , sl( z ) , sl( z ) , sl( z ) (cid:11) = (cid:10) sl(Ψ( z )) , sl(Ψ( z )) , sl(Ψ( z )) , sl(Ψ( z )) (cid:11) . (cid:3) Fact 2.3. [23, Lemma 2.20]
Let Λ ⊂ C . Then the cross ratio of slopes of fourpairwise nonparallel Λ -directions is an element of K + Λ . (cid:3) CHRISTIAN HUCK AND MICHAEL SPIEß Algebraic Delone sets
The following notions will be useful; see also [24], [26], [27] for generalisationsand for results related to those presented below.
Definition 3.1.
A Delone set Λ ⊂ C is called an algebraic Delone set if it satisfiesthe following properties:(Alg) [ K Λ : Q ] < ∞ . (Hom) For any finite subset F of K Λ , there is a homothety h of the complex plane such that h ( F ) ⊂ Λ .Moreover, Λ is called an n -cyclotomic Delone set if it satisfies the property( n -Cyc) K Λ ⊂ Q ( ζ n ) for some n ≥ and has property (Hom). Further, Λ is called a cyclotomic Deloneset if it is an n -cyclotomic Delone set for a suitable n ≥ . Remark 3.2.
Algebraic Delone sets were already introduced in [27, Definition 4.1].Clearly, for every algebraic Delone set Λ , the field extension K Λ / Q is an imaginaryextension (due to Λ being relatively dense) with K Λ = K Λ . By the Kronecker-Weber theorem (cf. [37, Thm. 14.1]) and Fact 2.1, the cyclotomic Delone sets areprecisely the algebraic Delone sets Λ with the additional property that K Λ / Q is anAbelian extension.Following Moody [29], modified along the lines of the algebraic setting of Pleas-ants [30], we define as follows. Definition 3.3.
Let K ⊂ C be an imaginary quadratic extension of a real al-gebraic number field (necessarily, this real algebraic number field is K + ) of de-gree [ K : Q ] =: d over Q (in particular, d is even). Let O K be the ring ofintegers in K and let . ⋆ : O K → C s − × R t be any map of the form z ( σ ( z ) , . . . , σ s ( z ) , σ s +1 ( z ) , . . . , σ s + t ( z )) , where σ s +1 , . . . , σ s + t are the real embed-dings of K/ Q into C / Q and σ , . . . , σ s arise from the complex embeddings of K/ Q into C / Q except the identity and the complex conjugation by choosing exactly oneembedding from each pair of complex conjugate ones (in particular, d = 2 s + t and s ≥ ). Then, for any such choice, each translate Λ of Λ ( W ) := { z ∈ O K | z ⋆ ∈ W } , where W ⊂ C s − × R t ≃ R d − is a relatively compact set with nonempty interior,is called a K -algebraic model set . Moreover, . ⋆ and W are called the star map andthe window of Λ , respectively. Remark 3.4.
Algebraic number fields K as above may be obtained by startingwith a real algebraic number field L and adjoining the square root of a negativenumber from L . Note that, in the situation of Definition 3.3, the quadratic extension K/K + is a Galois extension with G ( K/K + ) containing the identity and the complexconjugation (in particular, one has K = K ). We use the convention that for d = 2 (meaning that s = 1 and t = 0 ), C s − × R t is the trivial group { } and the starmap is the zero map. Due to the Minkowski representation { ( z, z ⋆ ) | z ∈ O K } ofthe maximal order O K of K being a (full) lattice in C × C s − × R t ≃ R d (cf. [10,Ch. 2, Sec. 3]) that is in one-to-one correspondence with O K via the canonicalprojection on the first factor and due to O ⋆K being a dense subset of C s − × R t (see Lemma 3.7 below), K -algebraic model sets are indeed model sets and thus areMeyer sets; cf. [8], [9], [29], [33], [34] for the general setting and further propertiesof model sets. Since the star map is a monomorphism of Abelian groups for d > and since the window is a bounded set, a K -algebraic model set Λ is periodic if andonly if d = 2 , in which case Λ is a translate of the planar lattice O K . NIQUENESS IN DISCRETE TOMOGRAPHY OF ALGEBRAIC DELONE SETS 5
A real algebraic integer λ is called a Pisot-Vijayaraghavan number ( PV-number )if λ > while all other conjugates of λ have moduli strictly less than . Fact 3.5. [32, Ch. 1, Thm. 2]
Every real algebraic number field contains a primitiveelement that is a PV-number. (cid:3)
Before we can show that K -algebraic model sets are algebraic Delone sets, weneed the following lemmata. Lemma 3.6.
Let Λ be a nonperiodic K -algebraic model set with star map . ⋆ . Then,there is an algebraic integer λ ∈ K + such that a suitable power of the Z -moduleendomorphism m ⋆λ of O ⋆K , defined by m ⋆λ ( z ⋆ ) = ( λz ) ⋆ , is contractive, i.e., there isan l ∈ N and a real number c ∈ (0 , such that k ( m ⋆λ ) l ( z ⋆ ) k ≤ c k z ⋆ k holds for all z ∈ O K .Proof. By Fact 3.5, we may choose a PV-number λ of degree d/ K + : Q ] in K + , where d = [ K : Q ] ≥ due to the nonperiodicity; see Remark 3.4. Since allnorms on C s − × R t ≃ R d − are equivalent, it suffices to prove the assertion in caseof the maximum norm on C s − × R t with respect to the absolute value on C and R ,respectively, rather than considering the Euclidean norm itself. But in that case,the assertion follows immediately with l := 1 and c := max (cid:8) | σ j ( λ ) | (cid:12)(cid:12) j ∈ { , . . . , s + t } (cid:9) , since the set { σ ( λ ) , . . . , σ s + t ( λ ) } of conjugates of λ does not contain λ itself. To seethis, note that σ j ( λ ) = λ , where j ∈ { , . . . , s + t } , implies that σ j fixes K + whence σ j is the identity or the complex conjugation, a contradiction; see Definition 3.3and Remark 3.4. (cid:3) Lemma 3.7.
Let Λ be a K -algebraic model set with star map . ⋆ and let d := [ K : Q ] .Then O ⋆K is dense in C s − × R t ≃ R d − .Proof. If d = 2 , one even has O ⋆K = C s − × R t = { } . Otherwise, choose a PV-number λ of degree d/ in K + ; cf. Fact 3.5. Since O K is a full Z -module in K , theset { λ k z | z ∈ O K } is a full Z -module in K for any k ∈ N . Thus the set { ( λ k z, ( m ⋆λ ) k ( z ⋆ )) | z ∈ O K } , is a (full) lattice in C s × R t ≃ R d for any k ∈ N , where m ⋆λ is the Z -moduleendomorphism of O ⋆K from Lemma 3.6; cf. [10, Ch. 2, Sec. 3]. In conjunction withLemma 3.6, this implies that, for any ε > , the Z -module O ⋆K contains an R -basisof C s − × R t whose elements have norms ≤ ε . The assertion follows. (cid:3) Lemma 3.8.
Let Λ be a K -algebraic model set. Then, for any finite set F ⊂ K ,there is a homothety h of the complex plane such that h ( F ) ⊂ Λ . Moreover, h canbe chosen such that h ( z ) = κz + v , where κ ∈ K + is an algebraic integer with κ ≥ and v ∈ Λ .Proof. Without loss of generality, we may assume that Λ is of the form Λ ( W ) (see Definition 3.3) and that F = ∅ . Note that there is an l ∈ N such that { lz | z ∈ F } ⊂ O K . Let d := [ K : Q ] and let . ⋆ be the star map of Λ . If d = 2 ,we are done by setting h ( z ) := lz . Otherwise, since W has nonempty interior,Lemma 3.7 shows the existence of a suitable z ∈ O K with z ⋆ ∈ W ◦ . Consider theopen neighbourhood V := W ◦ − z ⋆ of in C s − × R t and choose a PV-number λ of degree d/ in K + ; cf. Fact 3.5. By virtue of Lemma 3.6, there is a k ∈ N suchthat ( m ⋆λ ) k (cid:0) ( lF ) ⋆ (cid:1) ⊂ V .
It follows that { ( λ k z + z ) ⋆ | z ∈ lF } ⊂ W ◦ and, further, that h ( F ) ⊂ Λ , where h is the homothety given by z ( lλ k ) z + z . The additional statement followsimmediately from the observation that z ∈ Λ . (cid:3) CHRISTIAN HUCK AND MICHAEL SPIEß
Proposition 3.9. K -algebraic model sets are algebraic Delone sets. Moreover, any K -algebraic model set Λ satisfies K Λ = K .Proof. Since K Λ = K t + Λ for any t ∈ C , we may assume that Λ is of the form Λ ( W ) (see Definition 3.3). Any K -algebraic model set Λ is a Delone set by Remark 3.4.Property (Alg) follows from the observation that K Λ ⊂ K (recall that Λ − Λ ⊂O K and that K = K ). Further, property (Hom) is an immediate consequence ofLemma 3.8. Let { α , . . . , α d } be a Q -basis of K/ Q . By the additional statementof Lemma 3.8 there is a nonzero element κ ∈ K + and a point v ∈ Λ such thatthe Q -linear independent set { κα , . . . , κα d } is contained in Λ − { v } ⊂ K Λ . Since K Λ ⊂ K , this shows that K Λ = K . (cid:3) Remark 3.10.
As another immediate consequence of Lemma 3.8, one verifies that,for any K -algebraic model set Λ , the set of Λ -directions is precisely the set of O K -directions. Example 3.11.
Standard examples of n -cyclotomic Delone sets are the Q ( ζ n ) -algebraic model sets, where n ≥ , which from now on are called n -cyclotomicmodel sets ; cf. Fact 2.1 and Proposition 3.9 (note also that Q ( ζ n ) is obtained from Q ( ζ n ) + by adjoining the square root of the negative number ζ n + ζ − n − ∈ Q ( ζ n ) + ,the latter being the discriminant of X − ( ζ n + ζ − n ) X + 1 ). These sets were alsocalled cyclotomic model sets with underlying Z -module Z [ ζ n ] in [27, Sec. 4.5], since Z [ ζ n ] is the ring of integers in the n th cyclotomic field; cf. [37, Thm. 2.6]. The latterrange from periodic examples like the fourfold square lattice ( n = 4 ) or the sixfoldtriangular lattice ( n = 3 ) to nonperiodic examples like the vertex set of the tenfoldTübingen triangle tiling [6], [7] ( n = 5 ), the eightfold Ammann-Beenker tiling ofthe plane [1], [5], [19] ( n = 8 ) or the twelvefold shield tiling [19] ( n = 12 ); see [26,Fig. 1], [27, Fig. 2] and Fig. 1 below for illustrations. In general, for any divisor m of lcm( n, , one can choose the window such that the corresponding n -cyclotomicmodel sets have m -fold cyclic symmetry in the sense of symmetries of LI-classes,meaning that a discrete structure has a certain symmetry if the original and thetransformed structure are locally indistinguishable; cf. [2] for details. Note that thevertex sets of the famous Penrose tilings of the plane fail to be -cyclotomic modelsets but can still be seen to be -cyclotomic Delone sets; see [4] and referencestherein. 4. A cyclotomic theorem
Definition 4.1.
Let m ≥ be a natural number. Set D m := (cid:8) ( k , k , k , k ) ∈ N (cid:12)(cid:12) k < k ≤ k < k ≤ m − and k + k = k + k (cid:9) and define the function f m : D m → C ∗ by(4.1) f m ( k , k , k , k ) := (1 − ζ k m )(1 − ζ k m )(1 − ζ k m )(1 − ζ k m ) . We further set C m := f m ( D m ) (note that C m ⊂ C m ′ for any multiple m ′ of m )and C := S m ≥ C m . Moreover, for a subset K of C , we set C ( K ) := C ∩ K and C m ( K ) := C m ∩ K . Fact 4.2. [14, Lemma 3.1]
Let m ≥ . The function f m is real-valued. Moreover,one has f m ( d ) > for all d ∈ D m . (cid:3) For our application to discrete tomography, we shall below show the finiteness ofthe set C ( L ) for all real algebraic number fields L and provide explicit results in thethree cases Q ( ζ ) + = Q ( √ , Q ( ζ ) + = Q ( √ and Q ( ζ ) + = Q ( √ . Gardnerand Gritzmann showed the following result for the field Q = Q ( ζ ) + = Q ( ζ ) + . NIQUENESS IN DISCRETE TOMOGRAPHY OF ALGEBRAIC DELONE SETS 7
Theorem 4.3. [14, Lemma 3.8, Lemma 3.9 and Thm. 3.10] C ( Q ) = C ( Q ) = (cid:8) , , , , (cid:9) . Moreover, all solutions of f m ( d ) = q ∈ Q , where m ≥ and d ∈ D m , are eithergiven, up to multiplication of m and d by the same factor, by m = 12 and one ofthe following (i) d = (6 , , , , q = ; (ii) d = (6 , , , , q = 4; (iii) d = (4 , , , , q = ; (iv) d = (4 , , , , q = 3; (v) d = (4 , , , , q = ; (vi) d = (8 , , , , q = ; (vii) d = (4 , , , , q = 3; (viii) d = (8 , , , , q = 3; (ix) d = (3 , , , , q = 2; (x) d = (3 , , , , q = 2; (xi) d = (9 , , , , q = 2; or by one of the following (xii) d = (2 k, s, k, k + s ) , q = 2 , where s ≥ , m = 2 s and ≤ k ≤ s ; (xiii) d = ( s, k, k, k + s ) , q = 2 , where s ≥ , m = 2 s and s ≤ k < s. (cid:3) The next three lemmata are the key tools for our approach.
Lemma 4.4.
Let a ∈ R ∗ . If a = x y for x, y ∈ µ ∪ { } with y = − then a ∈ { , , } .Proof. It suffices to consider the cases a = 1 + ω and a = ω ω with ω, ω , ω ∈ µ and ω = − . In the first case, one has ω = a − ∈ µ ( R ) = {± } whence ω = 1 (due to a = 0 ) and a = 2 . In the second case, one has a = ¯ a = 1 + ¯ ω ω = ω ω − ω ω = ω ω − a wherefore ω = ω and a = 1 . (cid:3) Lemma 4.5 (Comparison of coefficients) . Let K ⊂ C be a field, let m ∈ N , and let ζ ∈ µ with ζ m ∈ K . Let a , . . . , a m − , b , . . . , b m − ∈ K with m − X i =0 a i ζ i = m − X i =0 b i ζ i . Then one has a i = b i for all i = 0 , . . . , m − if one of the following conditionsholds. (a) [ K ( ζ ) : K ] = m . (b) [ K ( ζ ) : K ] = m − and at most m − of a , . . . , a m − , b , . . . , b m − arenonzero.Moreover, if [ K ( ζ ) : K ] = m − and a k − b k = 0 for some k then | a i − b i | = | a j − b j | 6 = 0 for all i, j .Proof. In case (a), the assertion follows immediately from the linear independenceof , ζ, . . . , ζ m − over K . If [ K ( ζ ) : K ] = m − , set ω := ζ m ∈ K . The minimumpolynomial f ∈ K [ X ] of ζ over K has degree m − and one has X m − ω = ( X − ǫ ) f with ǫ ∈ K , hence ω = ǫ m (in particular, ǫ ∈ µ ( K ) ) and f = X m − ǫ m X − ǫ = m − X i =0 ǫ m − − i X i . If P m − i =0 ( a i − b i ) ζ i = 0 then there is an element c ∈ K with a i = b i + cǫ m − − i forall i = 0 , . . . , m − . By assumption (b) one has a i = 0 = b i for some i . This implies c = 0 and therefore the assertion. For the additional statement, first observe thatdue to a k = b k for some k one has c = 0 . Thus | a i − b i | = | cǫ m − − i | = | c | = | cǫ m − − j | = | a j − b j | 6 = 0 for all i, j . (cid:3) CHRISTIAN HUCK AND MICHAEL SPIEß
Lemma 4.6.
Let K ⊂ C be a field, let m ∈ N , and let ζ ∈ µ with ζ m ∈ K .Further, let ω , ω , ω , ω ∈ µ ( K ) and k , k , k , k ∈ { , . . . , m − } satisfy thefollowing conditions. • gcd( k i , m ) = 1 for some i ∈ { , , , } . • k + k ≡ k + k (mod m ) • ω ζ k , ω ζ k = 1 and a := (1 − ω ζ k )(1 − ω ζ k )(1 − ω ζ k )(1 − ω ζ k ) ∈ K ∩ ( R ∗ \ {± } ) .Then one has a ∈ { , } if one of the following conditions holds. (a) [ K ( ζ ) : K ] = m and m ≥ . (b) [ K ( ζ ) : K ] = m − and m ≥ .Proof. Without restriction, we may assume that gcd( k , m ) = 1 . Then, for i =2 , , , there are a i , b i ∈ Z such that k i = a i k + b i m and, with ζ ′ := ζ k , ζ k i =( ζ ′ ) a i ( ζ m ) b i . Since one has ( ζ ′ ) m ∈ K , K ( ζ ′ ) = K ( ζ ) and (1 − ω ζ k )(1 − ω ζ k )(1 − ω ζ k )(1 − ω ζ k ) = (1 − ω ′ ζ ′ )(1 − ω ′ ζ ′ k ′ )(1 − ω ′ ζ ′ k ′ )(1 − ω ′ ζ ′ k ′ ) for suitable ω ′ , ω ′ , ω ′ , ω ′ ∈ µ ( K ) and k ′ , k ′ , k ′ ∈ { , . . . , m − } with k ′ ≡ k ′ + k ′ (mod m ) , we may further assume that k = 1 . We thus obtain − ω ζ − ω ζ k + ω ω ζ k +1 = a − aω ζ k − aω ζ k + aω ω ζ k + k , where, without restriction, k ≤ k . From now on, let [ k ] ∈ { , . . . , m − } denotethe canonical representative of the equivalence class of k ∈ Z modulo m . We mayfinally write − ω ζ − ω ζ k + ω ω ωζ [ k +1] = a − aω ζ k − aω ζ [1+ k − k ] + aω ω ω ′ ζ [ k +1] with k ≤ [1 + k − k ] and suitable ω, ω ′ ∈ µ ( K ) . Case 1. k = 0 . Then − ω ζ − ω + ω ω ωζ = a − aω ζ k − aω ζ [1 − k ] + aω ω ω ′ ζ If k = 0 then a = − ω − ω by Lemma 4.5 and the assertion follows from Lemma 4.4.The case k = 1 cannot occur (due to k ≤ [1 − k ] ), whereas k ≥ implies a = 1 − ω by Lemma 4.5. The assertion follows from Lemma 4.4. Case 2. k = 1 . Then − ( ω + ω ) ζ + ω ω ωζ = a − aω ζ k − aω ζ [2 − k ] + aω ω ω ′ ζ If k = 0 then a = − ω by Lemma 4.5 and the assertion follows from Lemma 4.4.If k = 1 then Lemma 4.5 implies a = 1 , which is excluded by assumption. Thecase k = 2 is impossible (due to k ≤ [2 − k ] ). Let k ≥ (hence m ≥ ). Undercondition (a), this implies a = 1 by Lemma 4.5, which is excluded by assumption.Under condition (b), k = 3 implies − ( ω + ω ) ζ + ω ω ωζ = a − aω ζ − aω ζ m − + aω ω ω ′ ζ with m − ≥ (due to m ≥ ). The additional statement of Lemma 4.5 implies m = 5 and | − a | = | aω | = | a | , wherefore a = 1 / . If k ≥ then m ≥ (due to k ≤ [2 − k ] ) and Lemma 4.5 implies a = 1 , which is excluded by assumption. Case 3. k ∈ { , . . . , m − } (hence m ≥ and ≤ k < k + 1 ≤ m − ). Then (1 − a ) − ω ζ − ω ζ k + ( ω ω ω − aω ω ω ′ ) ζ k +1 = − aω ζ k − aω ζ [1+ k − k ] Under condition (a), Lemma 4.5 shows that this is impossible, since there are atleast three nontrivial coefficients on the left-hand side and at most two nontrivialcoefficients on the right-hand side of this equation. Under condition (b) (hence m ≥ ), k = 0 implies a − aω by Lemma 4.5 wherefore a = − ω and theassertion follows from Lemma 4.4. If k = 1 then a = 1 by Lemma 4.5, which is NIQUENESS IN DISCRETE TOMOGRAPHY OF ALGEBRAIC DELONE SETS 9 excluded by assumption. If k ≥ and m ≥ then a = 1 by Lemma 4.5, whichis excluded by assumption. Employing the additional statement of Lemma 4.5, weshall now see that the missing cases ( k ≥ and m ∈ { , } ) are either impossibleor yield | − a | = 1 and thus a = 2 (due to a = 0 ). In fact, m = 5 and k = 2 imply k = 3 (due to k ≤ [1 + k − k ] ) and, further, | − a | = | ω | = 1 . Thecase m = 5 and k = 3 cannot occur (due to k ≤ [1 + k − k ] ). If m = 5 and k = 4 then k = 2 (due to k ≤ [1 + k − k ] ) and, further, | − a | = | ω | = 1 . If m = 6 and k = 2 then k ∈ { , } (due to k ≤ [1 + k − k ] ). The case k = 3 is impossible, whereas the case k = 4 yields | − a | = | ω | = 1 . The case m = 6 and k = 3 is impossible (due to k ≤ [1 + k − k ] ). The case m = 6 and k = 4 implies k = 2 (due to k ≤ [1 + k − k ] ) and, further, | − a | = | ω | = 1 . Finally,the case m = 6 and k = 5 implies k = 3 (due to k ≤ [1 + k − k ] ) and, onceagain, | − a | = | ω | = 1 . Case 4. k = m − . Then (1 + ω ω ω ) − ω ζ − ω ζ m − = a (1 + ω ω ω ′ ) − aω ζ k − aω ζ [ m − k ] Under condition (a), Lemma 4.5 implies { k , [ m − k ] } = { , m − } , wherefore k = 1 and [ m − k ] = m − (due to k ≤ [ m − k ] ). Further, Lemma 4.5yields a = ω /ω , a contradiction (due to | a | 6 = 1 ). By the additional statementof Lemma 4.5, condition (b) (hence m ≥ ) implies m = 5 , k = 2 and, further, | aω | = | ω | = 1 , a contradiction (due to | a | 6 = 1 ). (cid:3) We are now in a position to prove the following extension of Theorem 4.3.
Theorem 4.7.
For n ∈ N , one has C ( Q ( ζ n ) + ) = C lcm(2 n, ( Q ( ζ n ) + ) . In particular, the last set is finite. Moreover, all solutions of f m ( d ) ∈ Q ( ζ n ) + ,where m ≥ and d ∈ D m , are either of the form (xii) or (xiii) of Theorem 4.3 orare given, up to multiplication of m and d by the same factor, by m = lcm(2 n, and d from a finite list.Proof. Since Q ( ζ n ) + = Q ( ζ n ) + for odd n it suffices to consider the case where n is even (hence lcm(2 n,
12) = lcm(2 n, ). Let m ≥ and d := ( k , k , k , k ) ∈ D m such that a := f m ( d ) = (1 − ζ k m )(1 − ζ k m )(1 − ζ k m )(1 − ζ k m ) ∈ Q ( ζ n ) + Recall that a > by Fact 4.2. We may assume that gcd( m, k , k , k , k ) = 1 . Byvirtue of Theorem 4.3, we may also assume that a Q . Observe that a ∈ Q ( ζ n ) + ∩ Q ( ζ m ) + = Q ( ζ gcd( m,n ) ) + Claims 1 and 2 below show that lcm(2 n, is a multiple of m , hence the assertion. Claim 1.
Let p be an odd prime number and assume that ord p ( n ) < ord p ( m ) .Then one has p = 3 , ord p ( m ) = 1 and ord p ( n ) = 0 .To see this, set r := ord p ( m ) ≥ , K := Q ( ζ m/p ) and note that Q ( ζ gcd( m,n ) ) ⊂ K .Let m = p r m ′ , where gcd( p, m ′ ) = 1 . Then, for i = 1 , , , , there are a i , b i ∈ Z such that k i = a i p + b i m ′ and, further, ζ k i m = ζ a i m/p ζ b i p r . Since ζ pp r = ζ p r − ∈ K onehas a = (1 − ω ζ l p r )(1 − ω ζ l p r )(1 − ω ζ l p r )(1 − ω ζ l p r ) for suitable ω , ω , ω , ω ∈ µ ( K ) and l , l , l , l ∈ { , . . . , p − } with gcd( l i , p ) = 1 for some i ∈ { , , , } and l + l ≡ l + l (mod p ) . Further, by Fact 2.1, one has [ K ( ζ p r ) : K ] = [ Q ( ζ m ) : Q ( ζ m/p )] = φ ( p r ) φ ( p r − ) = (cid:26) p − if r = 1 ; p if r ≥ . Lemma 4.6 implies both for p ≥ and r ≥ that a = 2 , a contradiction. Therefore p = 3 , r = ord p ( m ) = 1 and consequently ord p ( n ) = 0 . Claim 2. ord ( m ) ≤ ord ( n ) + 1 .Assume that r := ord ( m ) ≥ ord ( n ) + 2 ≥ . Set K := Q ( ζ m/ ) and note that Q ( ζ gcd( m,n ) ) ⊂ K . As above, since ζ r = ζ r − ∈ K , one has a = (1 − ω ζ l r )(1 − ω ζ l r )(1 − ω ζ l r )(1 − ω ζ l r ) for suitable ω , ω , ω , ω ∈ µ ( K ) and l , l , l , l ∈ { , , , } with gcd( l i ,
4) = 1 for some i ∈ { , , , } and l + l ≡ l + l (mod 4) . Further, by Fact 2.1, one has [ K ( ζ s ) : K ] = [ Q ( ζ m ) : Q ( ζ m/ )] = φ (2 r ) φ (2 r − ) = 4 . Lemma 4.6 now implies a = 2 , a contradiction. This proves the claim. (cid:3) Remark 4.8.
Similar to the proof of Theorem 4.7, one can also use Lemma 4.6 togive another proof of the fact shown in [14] that all solutions of f m ( d ) ∈ Q \ { } ,where m ≥ and d ∈ D m , are given, up to multiplication of m and d by the samefactor, by m = 12 . Thus the number plays a special role in this context. Indeedthis number leads to infinite families of solutions (see Theorem 4.3(xii)-(xiii) above)that can be found by using the -adic valuation; cf. [14] for details.One even has the following result, which improves [27, Thm. 4.19]. Theorem 4.9.
For any real algebraic number field L , the set C ( L ) is finite. More-over, there is a number m L ∈ N such that all solutions of f m ( d ) ∈ L , where m ≥ and d ∈ D m , are either of the form (xii) or (xiii) of Theorem 4.3 or are given, upto multiplication of m and d by the same factor, by m = m L and d from a finitelist.Proof. The finiteness of L/ Q together with the identity Q ( µ ) + = [ n ∈ N Q ( ζ n ) + implies that L ∩ Q ( µ ) + = L ∩ Q ( ζ n ) + for some n ∈ N . Since C ⊂ Q ( µ ) + by Fact 4.2it follows that C ( L ) = L ∩ C = L ∩ C ∩ Q ( µ ) + = L ∩ C ∩ Q ( ζ n ) + ⊂ C ( Q ( ζ n ) + ) . By virtue of Theorem 4.7, the assertion follows with m L := lcm(2 n, . (cid:3) Corollary 4.10. (a) C ( Q ( √ C ( Q ( √ (cid:8) − √ , √ , − √ , √ , √ , − √ , √ , , −√ , √ , , − √ , √ , √ , − √ , √ , , √ , √ , √ , √ , √ , , √ , √ , , √ , √ , √ , √ , √ , √ , √ (cid:9) Moreover, all solutions of f m ( d ) ∈ Q ( √ , where m ≥ and d ∈ D m ,are either of the form (xii) or (xiii) of Theorem 4.3 or are given, up tomultiplication of m and d by the same factor, by m = 60 and d from the NIQUENESS IN DISCRETE TOMOGRAPHY OF ALGEBRAIC DELONE SETS 11 following list. , , ,
42) 2 (24 , , ,
39) 3 (24 , , ,
54) 4 (36 , , , , , ,
30) 6 (36 , , ,
42) 7 (4 , , ,
10) 8 (5 , , , , , ,
44) 10 (8 , , ,
18) 11 (8 , , ,
35) 12 (8 , , , , , ,
25) 14 (9 , , ,
42) 15 (10 , , ,
16) 16 (10 , , , , , ,
54) 18 (12 , , ,
24) 19 (14 , , ,
32) 20 (14 , , , , , ,
46) 22 (16 , , ,
38) 23 (18 , , ,
28) 24 (18 , , , , , ,
42) 26 (18 , , ,
50) 27 (18 , , ,
56) 28 (21 , , , , , ,
36) 30 (24 , , ,
48) 31 (26 , , ,
42) 32 (26 , , , , , ,
44) 34 (28 , , ,
50) 35 (28 , , ,
55) 36 (32 , , , , , ,
50) 38 (34 , , ,
52) 39 (35 , , ,
57) 40 (36 , , , , , ,
55) 42 (42 , , ,
52) 43 (42 , , ,
54) 44 (46 , , , , , ,
56) 46 (52 , , ,
58) 47 (18 , , ,
30) 48 (42 , , , , , ,
54) 50 (5 , , ,
38) 51 (6 , , ,
22) 52 (8 , , , , , ,
33) 54 (8 , , ,
50) 55 (8 , , ,
58) 56 (9 , , , , , ,
44) 58 (10 , , ,
38) 59 (10 , , ,
54) 60 (12 , , , , , ,
28) 62 (14 , , ,
35) 63 (14 , , ,
42) 64 (14 , , , , , ,
38) 66 (18 , , ,
36) 67 (18 , , ,
44) 68 (18 , , , , , ,
54) 70 (21 , , ,
56) 71 (24 , , ,
48) 72 (25 , , , , , ,
44) 74 (26 , , ,
52) 75 (26 , , ,
54) 76 (28 , , , , , ,
50) 78 (32 , , ,
54) 79 (32 , , ,
57) 80 (34 , , , , , ,
54) 82 (39 , , ,
57) 83 (42 , , ,
56) 84 (42 , , , , , ,
58) 86 (12 , , ,
22) 87 (12 , , ,
33) 88 (12 , , , , , ,
54) 90 (24 , , ,
42) 91 (24 , , ,
50) 92 (24 , , , , , ,
54) 94 (36 , , ,
57) 95 (48 , , ,
58) 96 (8 , , , , , ,
30) 98 (18 , , ,
30) 99 (18 , , ,
45) 100 (32 , , , , , ,
50) 102 (36 , , ,
48) 103 (12 , , ,
30) 104 (24 , , , , , ,
54) 106 (15 , , ,
48) 107 (18 , , ,
36) 108 (30 , , , , , ,
40) 110 (10 , , ,
32) 111 (15 , , ,
21) 112 (18 , , , , , ,
36) 114 (30 , , ,
42) 115 (30 , , ,
46) 116 (30 , , , , , ,
51) 118 (15 , , ,
27) 119 (18 , , ,
42) 120 (30 , , , , , ,
57) 122 (8 , , ,
38) 123 (14 , , ,
44) 124 (18 , , , , , ,
48) 126 (24 , , ,
54) 127 (26 , , ,
56) 128 (28 , , , , , ,
57) 130 (10 , , ,
38) 131 (15 , , ,
54) 132 (18 , , , , , ,
48) 134 (30 , , ,
54) 135 (30 , , ,
56) 136 (30 , , , , , ,
54) 138 (30 , , ,
54) 139 (30 , , ,
42) 140 (12 , , , , , ,
28) 142 (12 , , ,
39) 143 (12 , , ,
50) 144 (24 , , , , , ,
42) 146 (24 , , ,
51) 147 (36 , , ,
46) 148 (36 , , , , , ,
54) 150 (14 , , ,
38) 151 (18 , , ,
54) 152 (34 , , , , , ,
36) 154 (18 , , ,
24) 155 (26 , , ,
48) 156 (42 , , , , , ,
48) 158 (20 , , ,
38) 159 (20 , , ,
54) 160 (40 , , , , , ,
32) 162 (40 , , ,
52) 163 (20 , , ,
26) 164 (20 , , , , , ,
46) 166 (20 , , ,
48) 167 (24 , , ,
44) 168 (36 , , , , , ,
48) 170 (30 , , ,
36) 171 (30 , , ,
40) 172 (30 , , , , , ,
45) 174 (20 , , ,
50) 175 (20 , , ,
30) 176 (40 , , , , , ,
35) 178 (40 , , ,
55) 179 (15 , , ,
50) 180 (15 , , , , , , (b) C ( Q ( √ C ( Q ( √ (cid:8) √ , − √ , √ , − √ , , √ , , √ , , , √ , , √ , , √ , √ , √ , √ (cid:9) . Moreover, all solutions of f m ( d ) ∈ Q ( √ , where m ≥ and d ∈ D m ,are either of the form (xii) or (xiii) of Theorem 4.3 or are given, up tomultiplication of m and d by the same factor, by m = 48 and d from thefollowing list. , , ,
20) 2 (10 , , ,
38) 3 (12 , , ,
18) 4 (12 , , , , , ,
33) 6 (12 , , ,
40) 7 (18 , , ,
26) 8 (18 , , , , , ,
39) 10 (24 , , ,
36) 11 (26 , , ,
40) 12 (30 , , , , , ,
44) 14 (36 , , ,
42) 15 (4 , , ,
12) 16 (8 , , , , , ,
36) 18 (10 , , ,
30) 19 (12 , , ,
24) 20 (15 , , , , , ,
36) 22 (20 , , ,
34) 23 (22 , , ,
36) 24 (22 , , , , , ,
42) 26 (38 , , ,
46) 27 (10 , , ,
24) 28 (18 , , , , , ,
40) 30 (30 , , ,
36) 31 (18 , , ,
32) 32 (4 , , , , , ,
14) 34 (9 , , ,
21) 35 (10 , , ,
28) 36 (10 , , , , , ,
36) 38 (18 , , ,
30) 39 (18 , , ,
42) 40 (20 , , , , , ,
40) 42 (26 , , ,
44) 43 (30 , , ,
42) 44 (33 , , , , , ,
46) 46 (6 , , ,
34) 47 (10 , , ,
20) 48 (12 , , , , , ,
34) 50 (12 , , ,
42) 51 (18 , , ,
36) 52 (18 , , , , , ,
46) 54 (22 , , ,
44) 55 (24 , , ,
42) 56 (30 , , , , , ,
46) 58 (10 , , ,
34) 59 (18 , , ,
42) 60 (22 , , , , , ,
34) 62 (24 , , ,
42) 63 (24 , , ,
46) 64 (12 , , , , , ,
30) 66 (24 , , ,
38) 67 (16 , , ,
22) 68 (32 , , , , , ,
34) 70 (30 , , ,
46) 71 (16 , , ,
42) 72 (24 , , , , , ,
40) 74 (16 , , ,
36) 75 (16 , , ,
40) 76 (16 , , , , , ,
40) 78 (16 , , ,
28) 79 (32 , , ,
44) 80 (12 , , , , , ,
20) 82 (36 , , , (c) C ( Q ( √ C ( Q ( √ (cid:8) − √ , √ , − √ , √ , √ , − √ , , √ , − √ , , √ , √ , √ , , √ , √ , √ , , √ , √ , , √ , √ , √ , √ , √ , √ , √ (cid:9) . Moreover, all solutions of f m ( d ) ∈ Q ( √ , where m ≥ and d ∈ D m ,are either of the form (xii) or (xiii) of Theorem 4.3 or are given, up tomultiplication of m and d by the same factor, by m = 24 and d from the NIQUENESS IN DISCRETE TOMOGRAPHY OF ALGEBRAIC DELONE SETS 13 following list. , , ,
10) 2 (6 , , ,
18) 3 (8 , , ,
14) 4 (8 , , , , , ,
20) 6 (10 , , ,
19) 7 (14 , , ,
20) 8 (16 , , , , , ,
16) 10 (4 , , ,
8) 11 (6 , , ,
16) 12 (9 , , , , , ,
16) 14 (10 , , ,
20) 15 (12 , , ,
18) 16 (18 , , , , , ,
16) 18 (4 , , ,
6) 19 (4 , , ,
15) 20 (5 , , , , , ,
18) 22 (6 , , ,
12) 23 (7 , , ,
20) 24 (8 , , , , , ,
14) 26 (10 , , ,
18) 27 (10 , , ,
21) 28 (14 , , , , , ,
20) 30 (17 , , ,
21) 31 (20 , , ,
22) 32 (4 , , , , , ,
18) 34 (14 , , ,
22) 35 (4 , , ,
18) 36 (6 , , , , , ,
15) 38 (8 , , ,
18) 39 (8 , , ,
22) 40 (10 , , , , , ,
21) 42 (16 , , ,
22) 43 (8 , , ,
12) 44 (14 , , , , , ,
11) 46 (4 , , ,
20) 47 (6 , , ,
14) 48 (6 , , , , , ,
18) 50 (12 , , ,
20) 51 (14 , , ,
22) 52 (15 , , , , , ,
13) 54 (4 , , ,
22) 55 (5 , , ,
11) 56 (5 , , , , , ,
18) 58 (7 , , ,
22) 59 (8 , , ,
16) 60 (10 , , , , , ,
22) 62 (14 , , ,
22) 63 (14 , , ,
23) 64 (17 , , , , , ,
13) 66 (6 , , ,
20) 67 (8 , , ,
18) 68 (8 , , , , , ,
22) 70 (16 , , ,
23) 71 (6 , , ,
10) 72 (6 , , , , , ,
14) 74 (8 , , ,
19) 75 (12 , , ,
18) 76 (16 , , , , , ,
14) 78 (12 , , ,
20) 79 (6 , , ,
22) 80 (10 , , , , , ,
22) 82 (8 , , ,
20) 83 (10 , , ,
22) 84 (8 , , , , , ,
22) 86 (12 , , ,
22) 87 (12 , , ,
14) 88 (4 , , , , , ,
16) 90 (10 , , ,
22) 91 (14 , , ,
20) 92 (6 , , , , , ,
22) 94 (8 , , ,
10) 95 (16 , , ,
18) 96 (10 , , , , , ,
22) 98 (10 , , ,
12) 99 (14 , , ,
16) 100 (12 , , , , , ,
20) 102 (8 , , ,
18) 103 (8 , , ,
20) 104 (8 , , , , , ,
20) 106 (8 , , ,
14) 107 (16 , , ,
22) 108 (6 , , , , , ,
10) 110 (18 , , , Proof.
Applying Theorem 4.7 to the cases n = 5 , , , the assertions follow froma direct computation. Note that in either case the last eleven entries of the listsabove derive from (i)-(xi) of Theorem 4.3. (cid:3) Determination of convex subsets of algebraic Delone sets byX-rays
Definition 5.1. (a) Let F be a finite subset of C , let u ∈ S be a direction,and let L u be the set of lines in the complex plane in direction u . Then the (discrete parallel) X-ray of F in direction u is the function X u F : L u → N := N ∪ { } , defined by X u F ( ℓ ) := card( F ∩ ℓ ) . (b) Let F be a collection of finite subsets of C and let U ⊂ S be a finite set ofdirections. We say that the elements of F are determined by the X-rays inthe directions of U if, for all F, F ′ ∈ F , one has ( X u F = X u F ′ ∀ u ∈ U ) ⇒ F = F ′ . The following negative result shows that, for algebraic Delone sets Λ , one hasto impose some restriction on the finite subsets of Λ to be determined. The proofonly needs property (Hom). Fact 5.2. [27, Prop. 3.1 and Remark 3.2]
Let Λ be an algebraic Delone set and let U ⊂ S be a finite set of pairwise nonparallel Λ -directions. Then the finite subsetsof Λ are not determined by the X-rays in the directions of U . (cid:3) Here, we shall focus on the convex subsets of algebraic Delone sets. One hasthe following fundamental result which even holds for Delone sets Λ with property(Hom). See Figure 1 for an illustration of direction (i) ⇒ (ii). Fact 5.3. [27, Prop. 4.6 and Lemma 4.5]
Let Λ be an algebraic Delone set and let U ⊂ S be a set of two or more pairwise nonparallel Λ -directions. The followingstatements are equivalent: (i) The convex subsets of Λ are determined by the X-rays in the directions of U . (ii) There is no U -polygon in Λ .In addition, if card( U ) < , then there is a U -polygon in Λ . (cid:3) The proof of the following central result uses Darboux’s theorem on secondmidpoint polygons; see [11], [17] or [13, Ch. 1].
Fact 5.4. [14, Prop. 4.2]
Let U ⊂ S be a finite set of directions. Then thereexists a U -polygon if and only if there is an affinely regular polygon such that eachdirection in U is parallel to one of its edges. (cid:3) Remark 5.5.
Clearly, U -polygons have an even number of vertices. Moreover,an affinely regular polygon with an even number of vertices is a U -polygon if andonly if each direction of U is parallel to one of its edges. On the other hand, it isimportant to note that a U -polygon need not be affinely regular, even if it is a U -polygon in an algebraic Delone set. For example, there is a U -icosagon in the vertexset of the Tübingen triangle tiling of the plane (a -cyclotomic model set; see [26,Fig. 1, Corollary 14 and Example 15]), which cannot be affinely regular since thatrestricts the number of vertices to , , , or by [24, Corollary 4.2]; see also [14,Example 4.3] for an example in the case of the square lattice. In general, there isan affinely regular polygon with n ≥ vertices in an algebraic Delone set Λ if andonly if Q ( ζ n ) + ⊂ K + Λ , the latter being a relation which (due to property (Alg)) canonly hold for finitely many values of n ; cf. [24, Thm. 3.3].We can now prove our main result on U -polygons which is an extension of [14,Thm. 4.5]. In fact, we use the same arguments as introduced by Gardner andGritzmann in conjunction with Fact 2.3 and Theorem 4.9. Note that the resulteven holds for arbitrary sets Λ with property (Alg). Theorem 5.6.
Let Λ be an algebraic Delone set. Further, let U ⊂ S be a setof four or more pairwise nonparallel Λ -directions and suppose the existence of a U -polygon. Then the cross ratio of slopes of any four directions of U , arranged inorder of increasing angle with the positive real axis, is an element of the set C ( K + Λ ) .Moreover, C ( K + Λ ) is finite and card( U ) is bounded above by a finite number b Λ ∈ N that only depends on Λ .Proof. Let U be as in the assertion. By Fact 5.4, U consists of directions parallelto the edges of an affinely regular polygon. There is thus a linear automorphism Ψ of the complex plane such that V := (cid:8) Ψ( u ) / | Ψ( u ) | (cid:12)(cid:12) u ∈ U (cid:9) is contained in a set of directions that are equally spaced in S , i.e., the anglebetween each pair of adjacent directions is the same. Since the directions of U arepairwise nonparallel, we may assume that there is an m ∈ N with m ≥ such thateach direction of V is given by e hπi/m , where h ∈ N satisfies h ≤ m − . Let u j , NIQUENESS IN DISCRETE TOMOGRAPHY OF ALGEBRAIC DELONE SETS 15 ≤ j ≤ , be four directions of U , arranged in order of increasing angle with thepositive real axis. By Fact 2.3, one has q := (cid:10) sl( u ) , sl( u ) , sl( u ) , sl( u ) (cid:11) ∈ K + Λ . We may assume that Ψ( u j ) / | Ψ( u j ) | = e h j πi/m , where h j ∈ N , ≤ j ≤ , and, h < h < h < h ≤ m − . Fact 2.2 now implies q = (cid:10) sl(Ψ( u )) , sl(Ψ( u )) , sl(Ψ( u )) , sl(Ψ( u )) (cid:11) = (tan( h πm ) − tan( h πm ))(tan( h πm ) − tan( h πm ))(tan( h πm ) − tan( h πm ))(tan( h πm ) − tan( h πm ))= sin( ( h − h ) πm ) sin( ( h − h ) πm )sin( ( h − h ) πm ) sin( ( h − h ) πm ) . Setting k := h − h , k := h − h , k := h − h and k := h − h , one gets ≤ k < k , k < k ≤ m − and k + k = k + k . Using sin( θ ) = − e − iθ (1 − e iθ ) / i ,one finally obtains K + Λ ∋ q = (1 − ζ k m )(1 − ζ k m )(1 − ζ k m )(1 − ζ k m ) = f m ( d ) , with d := ( k , k , k , k ) , as in (4.1). Then, d ∈ D m if its first two coordinates areinterchanged, if necessary, to ensure that k ≤ k ; note that this operation doesnot change the value of f m ( d ) . This proves the first assertion.Suppose that card( U ) ≥ . Let U ′ consist of seven directions of U and let V ′ := { Ψ( u ) / | Ψ( u ) | | u ∈ U ′ } . We may assume that all the directions of V ′ are inthe first two quadrants, so one of these quadrants, say the first, contains at leastfour directions of V ′ . Application of the above argument to these four directionsgives integers h j satisfying ≤ h < h < h < h ≤ m/ , where we may alsoassume, by rotating the directions of V ′ if necessary, that h = 0 . As above, weobtain a corresponding solution of f m ( d ) = q ∈ K + Λ , where d ∈ D m .By property (Alg) and Theorem 4.9, the set C ( K + Λ ) is finite and there is a number m Λ ∈ N such that all solutions of f m ( d ) ∈ K + Λ , where m ≥ and d ∈ D m , are eitherof the form (xii) or (xiii) of Theorem 4.3 or are given, up to multiplication of m and d by the same factor, by m = m Λ and d from a finite list. Without restriction,we may assume that m Λ is even.Suppose that the above solution is of the form (xii) or (xiii) of Theorem 4.3.Then using h = 0 , one obtains h = k = k + s > m/ , a contradiction. Thus,our solution derives from m = m Λ and finitely many values of d ∈ D m . Since thisapplies to any four directions of V ′ lying in the first quadrant, all such directionscorrespond to angles with the positive real axis which are integer multiples of π/m Λ .We claim that all directions of V ′ have the latter property. To see this, supposethat there is a direction v ∈ V ′ in the second quadrant, and consider a set offour directions v j , ≤ j ≤ , in V ′ , where v = v and v j , ≤ j ≤ , lie in thefirst quadrant. Suppose that v j = e h j πi/m , ≤ j ≤ . Then h j is an integermultiple of m/m Λ , for ≤ j ≤ . Again, we obtain a corresponding solutionof f m ( d ) = q ∈ K + Λ , where d ∈ D m . If this solution derives from the finite listguaranteed by Theorem 4.9, then clearly h is also an integer multiple of m/m Λ .Otherwise, by Theorem 4.9, this solution is of the form (xii) or (xiii) of Theorem 4.3and we can take h = 0 as before, whence either h = k , h = 2 k and h = k + s , ≤ k ≤ s/ , or h = s − k , h = s and h = k + s , s/ ≤ k < s , where m = 2 s .Since s = m/ m Λ / m/m Λ ) is an integer multiple of m/m Λ , we conclude ineither case that k , and hence h = k + s , is also an integer multiple of m/m Λ . Thisproves the claim. It thus remains to examine the case m = m Λ in more detail. Let h j , ≤ j ≤ ,correspond to the four directions of V ′ having the smallest angles with the positivereal axis, so that h = 0 and h j ≤ m/ , ≤ j ≤ . We have already shown thatthe corresponding d = ( k , k , k , k ) must occur in the finite list guaranteed byTheorem 4.9. Since h j ≤ m/ , ≤ j ≤ , we also have k j ≤ m/ , ≤ j ≤ . Thisyields only finitely many quadruples ( h , h , h , h ) = (0 , k − k , k , k ) .Suppose that h corresponds to any other direction of V ′ and replace ( h , h , h , h ) by ( h , h , h , h ) . We obtain finitely many d = ( h − h , h − h , h − h , h − h ) ∈ D m ,which, by Theorem 4.9, either occur in (xii) or (xiii) of Theorem 4.3 with m = m Λ oroccur in the finite list guaranteed by that result. This gives only finitely many pos-sible finite sets of more than four directions, which implies that card( U ) is boundedfrom above by a finite number that only depends on Λ (since the above analysisonly depends on Λ ). (cid:3) Similarly, the next result even holds for arbitrary sets Λ with property ( n -Cyc),where n ≥ . Theorem 5.7.
Let n ≥ and let Λ be an n -cyclotomic Delone set. Further, let U ⊂ S be a set of four or more pairwise nonparallel Λ -directions and suppose theexistence of a U -polygon. Then the cross ratio of slopes of any four directions of U , arranged in order of increasing angle with the positive real axis, is an elementof the subset C ( K + Λ ) of C ( Q ( ζ n ) + ) . Moreover C ( Q ( ζ n ) + ) = C lcm(2 n, ( Q ( ζ n ) + ) is finite and card( U ) is bounded above by a finite number b n ∈ N that only dependson n . In particular, one can choose b = b = 6 , b = 10 , b = 8 and b = 12 .Proof. Employing Theorem 4.7 together with the trivial observation that K + Λ ⊂ Q ( ζ n ) + for any n -cyclotomic Delone set, the general result follows from the samearguments as used in the proof of Theorem 5.6. The work of Gardner and Gritz-mann shows that one can choose b = b = 6 ; cf. [14, Thm. 4.5]. The specificbounds b n for n = 5 , , are obtained by following the proof of Theorem 5.6 andemploying Corollary 4.10.More precisely, let n = 8 (whence lcm(2 n,
12) = 48 ) and suppose that card( U ) ≥ . Let U ′ consist of seven directions of U and let V ′ := { Ψ( u ) / | Ψ( u ) | | u ∈ U ′ } , with Ψ as described in the proof of Theorem 5.6. Then all directions of V ′ correspond toangles with the positive real axis which are integer multiples of π/ and it sufficesto examine the case m = 48 in more detail. Let h j , ≤ j ≤ , correspond to thefour directions of V ′ having the smallest angles with the positive real axis, so that h = 0 and h j ≤ m/ , ≤ j ≤ . The corresponding d = ( k , k , k , k ) must occur in (1)-(82) of Corollary 4.10(b). Since h j ≤ , ≤ j ≤ , we also have k j ≤ , ≤ j ≤ . The only possibilities are (1), (3), (15), (19), (27), (28), (33),(34), (47), (67), (76) and (81) of Corollary 4.10(b). These yield ( h , h , h , h ) ∈ (cid:8) (0 , , , , (0 , , , , (0 , , , , (0 , , , , (0 , , , , (0 , , , , (0 , , , , (0 , , , , (0 , , , , (0 , , , , (0 , , , , (0 , , , (cid:9) . Suppose that h corresponds to any other direction of V ′ and replace ( h , h , h , h ) by ( h , h , h , h ) . The corresponding d either occur in (xii) or (xiii) of Theorem 4.3with m = 48 or occur in (1)-(82) of Corollary 4.10(b). We obtain (18 , h − , , h − , (12 , h − , , h − , (10 , h − , , h − , (18 , h − , , h − , (22 , h − , , h − , (18 , h − , , h − , (8 , h − , , h − , (15 , h − , , h − , (12 , h − , , h − , (16 , h − , , h − , (16 , h − , , h − and (12 , h − , , h − . The only possibilitiesare h = 24 , , , for (12 , h − , , h − , h = 26 , for (10 , h − , , h − , NIQUENESS IN DISCRETE TOMOGRAPHY OF ALGEBRAIC DELONE SETS 17 h = 30 , , for (18 , h − , , h − , h = 38 , for (22 , h − , , h − , h = 36 , for (18 , h − , , h − , h = 34 for (12 , h − , , h − , h = 32 , for (16 , h − , , h − and h = 34 for (12 , h − , , h − . It follows that theonly possible sets of more than four directions only comprise directions of the form e hπi/ and are given by the ranges { , , , , , } , { , , , , } , { , , , , , , , } , { , , , , , } , { , , , , , , } , { , , , , , } , { , , , , , } , { , , , , } of h . In particular, card( U ) ≤ .With the help of Corollary 4.10, the cases n = 5 , can be treated analogouslywith the following results.For n = 12 , the only possible sets of more than four directions only comprisedirections of the form e hπi/ and are given by the ranges { , , , , , , , } , { , , , , , , , , } , { , , , , , , , , } , { , , , , , , , , , } , { , , , , , , , , , , , } , { , , , , , , , } , { , , , , , , } , { , , , , , , , , } , { , , , , , , } , { , , , , , , , , } , { , , , , , , , } , { , , , , , } of h , whence card( U ) ≤ .For n = 5 , the only possible sets of more than four directions only comprisedirections of the form e hπi/ and are given by the ranges { , , , , , } , { , , , , , } , { , , , , , } , { , , , , , } , { , , , , } , { , , , , , , , , } , { , , , , } , { , , , , , , , } , { , , , , } , { , , , , } , { , , , , , } , { , , , , , } , { , , , , , , } , { , , , , , , , } , { , , , , , , , , , } , { , , , , , , , } of h , whence card( U ) ≤ in this case. (cid:3) Without further mention, the following result will be used in Remark 5.9 below.
Lemma 5.8.
Let Λ be a K -algebraic model set and let U ⊂ S be a finite set ofdirections. The following statements are equivalent: (i) There is a U -polygon in Λ . (ii) For any K -algebraic model set Λ ′ , there is a U -polygon in Λ ′ .Proof. The assertion follows immediately from Proposition 3.9 together with [27,Fact 4.4]. (cid:3)
Remark 5.9.
The work of Gardner and Gritzmann shows that b = b = 6 is bestpossible for any - or -cyclotomic model set; cf. [14, Example 4.3]. The U -icosagonin the vertex set of the Tübingen triangle tiling from Remark 5.5 has the propertythat card( U ) = 10 ; see [26, Figure 1]. This shows that, for any -cyclotomic modelset, the number b = 10 is best possible. Fig. 1 shows a U -polygon with verticesin the vertex set of the shield tiling with card( U ) = 12 , wherefore b = 12 isbest possible for any -cyclotomic model set. A similar example of a U -polygonwith vertices in the vertex set of the Ammann-Beenker tiling with card( U ) = 8 shows that b = 8 is best possible for any -cyclotomic model set; cf. [27, Fig. 2]. U -polygons of class c ≥ (i.e., U -polygons with consecutive edges parallel to Figure 1.
The boundary of a U -polygon in the vertex set Λ ofthe twelvefold shield tiling, where U is the set of twelve pairwisenonparallel Λ -directions given by the edges and diagonals of thecentral regular dodecagon. The vertices of Λ in the interior of the U -polygon together with the vertices indicated by the black andgrey dots, respectively, give two different convex subsets of Λ withthe same X-rays in the directions of U .directions of U ) in cyclotomic model sets were studied in [26]. By [26, Corollary14] (see also [12, Thm. 12]), the existence of a U -polygon of class c ≥ in an n -cyclotomic model set with n having the property that φ ( n ) / is equalto one or a prime number implies that card( U ) ≤ a n , where a = a = 6 , a = 8 , a = 12 and a n = 2 n for all other such values of n . In particular, one observes thecoincidence b n = a n for n = 3 , , , , ; cf. Theorem 5.7. However, there does notseem to be a reason why the least possible numbers b n in Theorem 5.7 may not belarger than a n for other n ≥ having the above property.Summing up, we finally obtain our main result on the determination of convexsubsets of algebraic Delone sets; see [27, Thm. 4.21] for a weaker version. Theorem 5.10.
Let Λ be an algebraic Delone set. (a) There are sets of four pairwise nonparallel Λ -directions such that the convexsubsets of Λ are determined by the corresponding X-rays. In addition, lessthan four pairwise nonparallel Λ -directions never suffice for this purpose. (b) There is a finite number c Λ ∈ N such that the convex subsets of Λ aredetermined by the X-rays in any set of c Λ pairwise nonparallel Λ -directions.Proof. To prove (a), it suffices by Fact 5.3 and Theorem 5.6 to take any set of fourpairwise nonparallel Λ -directions such that the cross ratio of their slopes, arrangedin order of increasing angle with the positive real axis, is not an element of the finiteset C ( K + Λ ) . Since Λ is relatively dense, the set of Λ -directions is dense in S . Inparticular, this shows that the set of slopes of Λ -directions is infinite. For exampleby fixing three pairwise nonparallel Λ -directions and letting the fourth one vary, onesees from this that the set of cross ratios of slopes of four pairwise nonparallel Λ -directions, arranged in order of increasing angle with the positive real axis, is infinite NIQUENESS IN DISCRETE TOMOGRAPHY OF ALGEBRAIC DELONE SETS 19 as well. The assertion follows. The additional statement follows immediately fromFact 5.3. Part (b) is a direct consequence of Fact 5.3 and Theorem 5.6. (cid:3)
The following result improves [27, Thm. 4.33] and particularly solves Problem4.34 of [27]; cf. Example 3.11 and compare [14, Thm. 5.7].
Theorem 5.11.
Let n ≥ and let Λ be an n -cyclotomic Delone set. (a) There are sets of four pairwise nonparallel Λ -directions such that the convexsubsets of Λ are determined by the corresponding X-rays. In addition, lessthan four pairwise nonparallel Λ -directions never suffice for this purpose. (b) There is a finite number c n ∈ N that only depends on n such that theconvex subsets of Λ are determined by the X-rays in any set of c n pairwisenonparallel Λ -directions. In particular, one can choose c = c = 7 , c =11 , c = 9 and c = 13 .Proof. Part (a) follows immediately from Theorem 5.10(a). Note that, by Fact 5.3and Theorem 5.7, it suffices to take any set of four pairwise nonparallel Λ -directionssuch that the cross ratio of their slopes, arranged in order of increasing angle withthe positive real axis, is not an element of the finite set C ( Q ( ζ n ) + ) . Part (b) is adirect consequence of Fact 5.3 in conjunction with Theorem 5.7. (cid:3) Remark 5.12.
Remark 5.9 shows that, for any n -cyclotomic model set with n =3 , , , , , the number c n above is best possible with respect to the numbers ofX-rays used. As already explained in the introduction, for practical applications,one additionally has to make sure that the Λ -directions used yield densely occupiedlines in Λ . For the practically most relevant case of n -cyclotomic model sets with n = 3 , , , , , this can actually be achieved; cf. [14, Remark 5.8] and [27, Sec. 4]for examples of suitable sets of four pairwise nonparallel Λ -directions in these cases.For the latter examples also recall that, for any n -cyclotomic model set Λ , theset of Λ -directions is precisely the set of Z [ ζ n ] -directions; cf. Remark 3.10 andExample 3.11. It was shown in [25, Prop. 3.11] that icosahedral model sets Λ ⊂ R can be sliced orthogonal to a fivefold axis of their underlying Z -module into -cyclotomic model sets. Applying Theorem 5.11 to each such slice, one sees that theconvex subsets of Λ are determined by the X-rays in suitable four and any elevenpairwise nonparallel Λ -directions orthogonal to the slicing axis.6. Determination of convex bodies by continuous X-rays
In [17], the following continuous version of Fact 5.3 was shown; compare Fact 5.4.Here, the continuous X-ray of a convex body K ⊂ C (i.e., K is compact withnonempty interior) in direction u ∈ S gives the length of each chord of K parallelto u and the concept of determination is defined as in the discrete case; cf. [13],[17] for details. Fact 6.1.
Let U ⊂ S be a set of two or more pairwise nonparallel directions. Thefollowing statements are equivalent: (i) The convex bodies in C are determined by the continuous X-rays in thedirections of U . (ii) There is no U -polygon.In addition, if card( U ) < , then there is a U -polygon. (cid:3) Employing Fact 6.1 instead of Fact 5.3, the following result follows from thesame arguments as used in the proofs of Theorems 5.10 and 5.11; compare [14,Thm. 6.2]. Note that neither the uniform discreteness of Λ nor property (Hom) areneeded in the proof. More precisely, our proof of part (a) needs property (Alg) and the relative denseness of Λ , whereas part (b) and the additional statement hold forarbitrary sets Λ with property (Alg) and ( n -Cyc) (where n ≥ ), respectively. Theorem 6.2.
Let Λ be an algebraic Delone set. (a) There are sets of four pairwise nonparallel Λ -directions such that the convexbodies in C are determined by the corresponding continuous X-rays. Inaddition, less than four pairwise nonparallel Λ -directions never suffice forthis purpose. (b) There is a finite number c Λ ∈ N such that the convex bodies in C aredetermined by the continuous X-rays in any set of c Λ pairwise nonparallel Λ -directions.Moreover, for any n -cyclotomic Delone set Λ , there is a finite number c n ∈ N thatonly depends on n such that the convex bodies in C are determined by the continuousX-rays in any set of c n pairwise nonparallel Λ -directions. In particular, one canchoose c = c = 7 , c = 11 , c = 9 and c = 13 . (cid:3) Remark 6.3.
Employing the U -polygons from Remark 5.9, it is straightforwardto show that the above numbers c n , where n = 3 , , , , , are best possible. Acknowledgements
This work was supported by the German Research Council (Deutsche Forschungs-gemeinschaft), within the CRC 701. C. H. is grateful to Richard J. Gardner for hiscooperation and encouragement. The authors thank M. Baake for useful commentson the manuscript.
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