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THE COINCIDENCE PROBLEM FOR SHIFTED LATTICES AND MULTILATTICES MANUEL JOSEPH C. LOQUIASAND PETER ZEINER 3 1 0 Abstract. Acoincidencesitelatticeisasublatticeformedbytheintersectionofalattice 2 Γin d with theimage of Γunderalinear isometry. Suchalinearisometry is referred to R n asalinearcoincidenceisometryofΓ. Here,weconsiderthemoregeneralcaseallowingany a affine isometry. Consequently, general results on coincidence isometries of shifted copies J of lattices, and of multilattices are obtained. In particular, we discuss the shifted square 6 lattice and thediamond packingin detail. 1 ] G M 1. Introduction and Outline . h ItwasFriedelin1911whofirstrecognizedtheusefulnessofcoincidencesitelattices(CSLs) t a in describing and classifying grain boundaries of crystals [11]. Since then, CSLs have been m an indispensable tool in the study of grain boundaries, twins, and interfaces [24, 6, 35]. [ This prompted various authors to examine the CSLs of cubic and hexagonal crystals [30, 17, 14, 18]. 1 v Theadvent of quasicrystals in 1984 triggered a renewed interest in CSLs. Thisis because 9 experimental evidence showed that quasicrystals, like ordinary crystals, exhibit multiple 8 grains, twin relationships, and coincidence quasilattices [34, 36]. This led to a more general 6 3 and mathematical treatment of the coincidence problem for lattices in [1]. . Various results are now known about the coincidences of lattices and modules in di- 1 0 mensions at most four. The coincidence problem for certain planar lattices and modules 3 was solved in [28, 1] using factorization properties of cyclotomic integers. For lattices 1 and modules in dimensions three and four, quaternions have proven to be an appropriate : v tool [1, 38, 4, 31, 39, 5, 2, 20, 21]. i X However, the mathematical treatment of the coincidence problem has been mostly re- r stricted to linear coincidence isometries, whereas isometries containing a translational part a are usually ignored. Nevertheless, general (affine) isometries are important in crystallogra- phy. Indeed, the situation where one shifts the two component crystals against each other was investigated in [12, 10] and references therein. Even though the idea of introducing a shift after applying a linear coincidence isometry has already been dealt with in the physical literature, not much can be found in the mathematical literature where a systematic treatment of the subject is still missing. Initial steps in this general direction have actually been made in the appendix of [28]. There, the authors considered coincidence isometries about certain points that are not lattice or Date: January 17, 2013. 2010 Mathematics Subject Classification. Primary 52C07; Secondary 11H06, 82D25, 52C23. Key words and phrases. coincidence isometry, coincidence site lattice, grain boundary, multilattice, diamond lattice. 1 2 M.J.C.LOQUIASANDP.ZEINER module points. For example, they determined the set of coincidence isometries about the center of a Delauney cell of the square lattice and calculated the corresponding indices. In this paper, the notion of a CSL is extended to intersections of two lattices that are related by any isometry. Such intersections are referred to as affine coincidence site lattices (ACSLs), and the isometries that generate these intersections as affine coincidence isometries. Theorem 3.3 identifies the affine coincidence isometries of a lattice, while Equa- tion (3.1) gives the resulting intersections. The succeeding discussion covers a related and special case: the coincidence problem for shifted lattices. That is, after translating the lattice Γ by some vector x, and upon application of a linear isometry R to the shifted lattice x+Γ, its intersection with x+Γ is considered. Theorem 4.3 asserts that the linear coincidence isometries of x+Γ are precisely thosecoincidence isometries R of Γthatsatisfy Rx x Γ+RΓ. Moreover, theCSLsof the − ∈ shifted lattice are merely translates of CSLs of the original lattice. Hence, no new values of coincidence indices are obtained by shifting the lattice, with some values disappearing or their multiplicity being changed. Similar to the approach in [28, 1], an extensive analysis of the coincidences of a shifted square lattice in Section 5 is achieved by identifying the square lattice with the ring of Gaussian integers. Thecoincidence problem for a shifted square lattice is completely solved when the shift consists of an irrational component (Theorem 5.9). For the remaining case, that is, when the shift may be written as a quotient of two Gaussian integers that are relatively prime, one can compute for the set of coincidence rotations of the shifted square lattice using a divisibility condition involving the denominator of the shift (Lemma 5.11). In both instances, the set of coincidence rotations of a shifted square lattice forms a group. An example is given where the set of coincidence isometries of a shifted square lattice is not a group. The latter part of this contribution is concerned with the coincidences of sets of points formed by the union of a lattice with a finite number of shifted copies of the lattice. Such sets are referred to as multilattices (see [27] and references therein). This idea should be useful for crystals having multiple atoms per primitive unit cell [13, 29]. Theorem 6.3 gives the solution of the coincidence problem for multilattices. Simply put, the linear coincidence isometriesofamultilatticeareexactlythecoincidenceisometriesofthelatticethatgenerates the multilattice - only the resulting intersections and corresponding indices may vary. This paves the way for the solution of the coincidence problem for the diamond packing given in Theorem 7.3. 2. Linear coincidences of lattices We start with the basic definitions and some known results on linear coincidence isometries of lattices. A discrete subset Γ of d is a lattice (of rank and dimension d) if it is the -span of R Z d linearly independent vectors v ,...,v d over . The set v ,...,v is called a basis 1 d 1 d ∈ R R { } of Γ, and Γ = v v . As a group, Γ is isomorphic to the free abelian group of 1 d Z ⊕···⊕Z rankd. Alternatively, onecancharacterize alatticein d asadiscreteco-compact subgroup R of d. A subset Γ′ of Γ is a sublattice of Γ if Γ′ is a subgroup of Γ of finite (group) index. R Hence, Γ′ is itself a lattice of the same rank and dimension as Γ. The index of Γ′ in Γ may be interpreted geometrically – [Γ : Γ′] is the quotient of the volume of a fundamental domain of Γ′ by the volume of a fundamental domain of Γ. THE COINCIDENCE PROBLEM FOR SHIFTED LATTICES AND MULTILATTICES 3 For a lattice Γ in d, its dual lattice or reciprocal lattice Γ∗ is defined by R Γ∗ := x d : x,y for all y Γ , ∈ R h i ∈ Z ∈ n o where , denotes the standard scalar product in d. Given a sublattice Γ′ of Γ, Γ∗ is a h· ·i R sublattice of (Γ′)∗ with [(Γ′)∗ :Γ∗] = [Γ : Γ′] and (Γ′)∗/Γ∗ = Γ/Γ′ [1, Lemma 2.3]. ∼ Two lattices Γ and Γ are said to be commensurate, denoted Γ Γ , if Γ Γ is 1 2 1 2 1 2 ∼ ∩ a sublattice of both Γ and Γ . Commensurateness between lattices defines an equiva- 1 2 lence relation [1, Proposition 2.1]. Given two commensurate lattices Γ and Γ , their sum 1 2 Γ +Γ := x +x :x Γ ,x Γ is also a lattice. In fact, the following equations 1 2 1 2 1 1 2 2 hold: (Γ Γ{ )∗ = Γ∗+Γ∈∗ and (Γ∈+Γ} )∗ = Γ∗ Γ∗ [1, Proposition 2.2]. 1∩ 2 1 2 1 2 1∩ 2 AnorthogonaltransformationR O(d) := O(d, )isalinearcoincidenceisometry ofthe ∈ R latticeΓin d ifΓ RΓ. ThesublatticeΓ(R):= Γ RΓiscalled thecoincidence site lattice R ∼ ∩ (CSL)ofΓgeneratedbyR,whiletheindexofΓ(R)inΓ,Σ (R) := [Γ : Γ(R)] = [RΓ :Γ(R)], Γ is referred to as the coincidence index of R with respect to Γ. If no confusion arises, we simply write Σ(R) to denote the coincidence index of R. Clearly, symmetries in the point group of Γ, P(Γ) = R O(d) :RΓ = Γ , are precisely those linear coincidence isometries { ∈ } R of Γ with Σ(R)= 1. The set of linear coincidence isometries of a lattice Γ in d is denoted by OC(Γ) while R the set of coincidence rotations of Γ, that is, OC(Γ) SO(d), is written as SOC(Γ). Since ∩ commensuratenessoflattices isanequivalencerelation, thesetOC(Γ)formsagrouphaving SOC(Γ) as a subgroup [1, Theorem 2.1]. 3. Affine coincidences of lattices Let Γ be a lattice in d. A subset of Γ will be called a cosublattice of Γ if it is a coset R ℓ + Γ′ of some sublattice Γ′ of Γ. The index of a cosublattice ℓ + Γ′ of Γ, denoted by [Γ : ℓ+Γ′], is defined as the index of the sublattice Γ′ in Γ. This definition of index makes sense geometrically: a translation does not change the volume of the fundamental domains of Γ and Γ′. Denote by E(d) the group of isometries of d. An element of E(d) shall be written as R (v,R), where (v,R) : x v+R(x), with R O(d) (the linear part of f) and v d (the 7→ ∈ ∈ R translational partof f). Thedefinition below generalizes theconcept of a linear coincidence isometry to an affine coincidence isometry. Definition 3.1. Let Γ be a lattice in d and (v,R) E(d). Then (v,R) is an affine R ∈ coincidence isometry of Γ if Γ (v,R)Γ contains a cosublattice of Γ. ∩ The set of affine coincidence isometries of Γ shall be denoted by AC(Γ). It is easy to see that AC(Γ) contains the group OC(Γ) = AC(Γ) O(d) = (v,R) AC(Γ):v = 0 . ∩ { ∈ } The following lemma describes the intersection of two lattices that are related by some isometry. Lemma 3.2. Let Γ d be a lattice and (v,R) E(d). If v ℓ+RΓ for some ℓ Γ, ⊆ R ∈ ∈ ∈ then Γ (v,R)Γ = ℓ+(Γ RΓ). ∩ ∩ Proof. Since v ℓ+RΓ, we have (v,R)Γ = (ℓ,R)Γ. It remains to show that Γ (ℓ,R)Γ = ∈ ∩ ℓ+(Γ RΓ). Takex Γ (ℓ,R)Γandwritex = ℓ+Rℓ′ forsomeℓ′ Γ. Thenx ℓ+(Γ RΓ) becau∩se Rℓ′ =x ℓ∈ Γ∩ RΓ. The opposite inclusion is clear. ∈ ∈ ∩ (cid:3) − ∈ ∩ Lemma 3.2 brings about the following characterization of an affine coincidence isometry of a lattice. 4 M.J.C.LOQUIASANDP.ZEINER Theorem 3.3. Let Γ be a lattice in d. Then (v,R) E(d) is an affine coincidence R ∈ isometry of Γ if and only if R OC(Γ) and v Γ+RΓ. ∈ ∈ Proof. It follows from Lemma 3.2 that if R OC(Γ) and v Γ+RΓ then Γ (v,R)Γ is a ∈ ∈ ∩ coset of Γ(R). Hence, (v,R) E(d) is an affine coincidence isometry of Γ. ∈ In the other direction, let (v,R) AC(Γ). Because Γ (v,R)Γ = ∅, there exist ℓ,ℓ′ Γ ∈ ∩ 6 ∈ with ℓ = v+Rℓ′, and so v = ℓ Rℓ′ Γ+RΓ. By Lemma 3.2, one obtains [Γ :Γ RΓ]= [Γ : ℓ+(Γ RΓ)]= [Γ :Γ (v,−R)Γ] <∈ . This implies that Γ RΓ and R OC(∩Γ). (cid:3) ∩ ∩ ∞ ∼ ∈ Therefore, the set of affine coincidence isometries of Γ is given by AC(Γ)= (v,R) E(d) : R OC(Γ) and v Γ+RΓ . { ∈ ∈ ∈ } Moreover, if (v,R) AC(Γ) with v ℓ+RΓ for some ℓ Γ, then ∈ ∈ ∈ (3.1) Γ (v,R)Γ = ℓ+Γ(R) ∩ by Lemma 3.2. Thus, Γ (v,R)Γ is a coset of Γ(R). This means that the intersection ∩ Γ (v,R)Γ does not only contain a cosublattice of Γ but is in fact a cosublattice of Γ. For ∩ this reason, we shall refer to Γ (v,R)Γ as an affine coincidence site lattice (ACSL) of Γ. ∩ In addition, each R OC(Γ) corresponds to Σ(R) distinct possible ACSLs. ∈ Remark3.4. Anotherlatticeofinterestinthestudyofgrainboundariesisthedisplacement shift complete (DSC) lattice. It is the lattice formed by all possible displacement vectors that preserve the structureof the grain boundary. In this setting, given a linear coincidence isometry R ofthelattice Γ, thecorrespondingDSClattice is v :(v,R) AC(Γ) = Γ+RΓ { ∈ } by Theorem 3.3. This conclusion is in agreement with the main result of [15], which states that the DSC lattice generated by R is the dual lattice of the CSL of Γ∗ obtained from R, that is, (Γ∗ RΓ∗)∗ = Γ+RΓ. ∩ Now, the identity isometry AC(Γ) for any lattice Γ in d. In addition, it follows d 1 ∈ R from Theorem 3.3 that the inverse of every isometry in AC(Γ) is also in AC(Γ). However, the product of two affine coincidence isometries of Γ may or may not be an element of AC(Γ). Thus, the set AC(Γ) does not always form a group. The next proposition tells us exactly when AC(Γ) is a group. Proposition 3.5. Let Γ d be a lattice. Then AC(Γ) is a group if and only if it is the ⊆ R symmetry group G of Γ. Proof. Suppose AC(Γ) is a group and take (v,R) AC(Γ). By Theorem 3.3, R−1 ∈ ∈ OC(Γ) AC(Γ). Thus, the product (v,R)(0,R−1) = (v, ) AC(Γ). It follows then d ⊆ 1 ∈ from Theorem 3.3 that v Γ. Furthermore, Γ+RΓ = Γ and hence, R P(Γ). Since G ∈ ∈ is symmorphic, that is, G is the semidirect product of P(Γ) with its translation subgroup T(G) = Γ, one obtains (v,R) G. (cid:3) ∈ 4. Linear coincidences of shifted lattices We now turn our attention to shifted copies x + Γ of a lattice Γ in d obtained by R translating all the points of Γ by the vector x d. By a cosublattice of the shifted lattice ∈ R x+Γ, we mean a subset of x+Γ of the form x+(ℓ+Γ′) where ℓ+Γ′ is a cosublattice of Γ. In addition, the index of the cosublattice x+(ℓ+Γ′) in x+Γ is understood to be [x+Γ :x+(ℓ+Γ′)] := [Γ : Γ′]. There is no ambiguity here - relabeling x as the origin gives back the original lattice Γ and cosublattice ℓ+Γ′. Of particular interest in this section are intersections of the form (x+Γ) R(x+Γ), where R O(d). ∩ ∈ THE COINCIDENCE PROBLEM FOR SHIFTED LATTICES AND MULTILATTICES 5 Definition 4.1. Let Γ be a lattice in d and x d. An R O(d) is said to be a linear R ∈ R ∈ coincidence isometry of the shifted lattice x+Γ if (x+Γ) R(x+Γ) is a cosublattice of ∩ x+Γ. The intersection (x + Γ) R(x + Γ) will also be referred to as a CSL of the shifted ∩ lattice x+Γ. The coincidence index of R with respect to x+Γ is taken to be Σ (R) := x+Γ [x+Γ : (x+Γ) R(x+Γ)]. The set of all linear coincidence isometries of x+Γ shall be ∩ denoted by OC(x+Γ). Likewise, we take SOC(x+Γ):= OC(x+Γ) SO(d). ∩ Remark 4.2. Observe that applying a linear isometry R on the shifted lattice x + Γ is equivalent to applying the same isometry R but with center at x on the original lattice − Γ. Hence, just as OC(Γ) is an extension of P(Γ), one may interpret OC(x + Γ) as a generalization of the stabilizer of the point x. − The following theorem characterizes a linear coincidence isometry R of a shifted lattice x+Γ and identifies the CSL of x+Γ generated by R. Theresult lies on the fact that taking the intersection of x+Γ and R(x+Γ) corresponds to a shift of the intersection of Γ and (Rx x,R)Γ by x. It is a special case of Lemma 6.1 which will be stated and proved in − Section 6. Theorem 4.3. Let Γ be a lattice in d and x d. Then R ∈ R OC(x+Γ)= R OC(Γ): Rx x Γ+RΓ . { ∈ − ∈ } In addition, if R OC(x+Γ) with Rx x ℓ+RΓ for some ℓ Γ, then ∈ − ∈ ∈ (4.1) (x+Γ) R(x+Γ)= (x+ℓ)+Γ(R). ∩ Equation (4.1) indicates that the CSL of the shifted lattice x + Γ generated by R ∈ OC(x+Γ) is obtained by translating some coset of Γ(R) in Γ by x. Consequently, (4.2) Σ (R) = Σ (R) x+Γ Γ for all R OC(x+Γ). This means that shifting a lattice does not yield any new values of ∈ coincidence indices. Let S P(Γ). If R OC(Γ) then RS OC(Γ) and the CSLs generated by R and RS ∈ ∈ ∈ are the same, that is, Γ(RS) = Γ(R). The corresponding statement for linear coincidence isometries of shifted lattices reads as follows. It will prove to be useful when counting the number of CSLs of a shifted lattice for a given index. Proposition 4.4. Let x+Γ d be a shifted lattice, S P(Γ), and suppose that R,RS ⊆ R ∈ ∈ OC(x+Γ). Then (x+Γ) RS(x+Γ) = (x+Γ) R(x+Γ) if and only if S OC(x+Γ). ∩ ∩ ∈ In particular, if OC(x+Γ) forms a group, then (x+Γ) RS(x+Γ)= (x+Γ) R(x+Γ). ∩ ∩ Proof. It follows from Theorem 4.3 that Rx x ℓ +RΓ and RSx x ℓ +RΓ for some 1 2 − ∈ − ∈ ℓ ,ℓ Γ. Equation (4.1) yields that (x+Γ) RS(x+Γ)= (x+Γ) R(x+Γ) if and only 1 2 ∈ ∩ ∩ if ℓ ℓ RΓ. However, R(Sx x) = RSx Rx (ℓ ℓ )+ RΓ. This implies that 2 1 2 1 ℓ ℓ− R∈Γ if and only if Sx x− Γ. Applyin−g The∈orem 4−.3 proves the claim. (cid:3) 2 1 − ∈ − ∈ Note that for an S P(Γ), the condition S OC(x+Γ) in Proposition 4.4 is equivalent ∈ ∈ to saying that S is an element of the stabilizer of x (see Remark 4.2). − Proposition 4.5. Let Γ d be a lattice and x d. If S P(Γ) then ⊆ R ∈ R ∈ OC(Sx+Γ)= S[OC(x+Γ)]S−1. 6 M.J.C.LOQUIASANDP.ZEINER Proof. This is a consequence of Theorem 4.3 because SRS−1(Sx) Sx Γ+SRS−1Γ if and only if Rx x Γ+RΓ for all R OC(Γ). − ∈ (cid:3) − ∈ ∈ For a given lattice Γ d, it is enough to consider values of x in a fundamental domain ⊆ R of Γ to compute for all the different possible sets OC(x+Γ). Proposition 4.5 asserts even more: it suffices to look at values of x in a fundamental domain of the symmetry group of Γ. Furthermore, the following inclusion property follows immediately from Theorem 4.3. Lemma 4.6. If Γ is a lattice in d and x,y d, then for all a,b , R ∈ R ∈Z OC(x+Γ) OC(y+Γ) OC[(ax+by)+Γ]. ∩ ⊆ Corollary 4.7. Let Γ be a lattice in d and x = 1ℓ, where ℓ Γ and n . If a R n ∈ ∈ N ∈ Z with a and n relatively prime, then OC(ax+Γ)= OC(x+Γ). Proof. The inclusion OC(x+Γ) OC(ax+Γ) follows directly from Lemma 4.6. Since a ⊆ is relatively prime to n, there exist integers b and c such that ab+nc= 1. Applying again Lemma 4.6 yields OC(ax+Γ) OC[(ab+nc)1ℓ+Γ]= OC(x+Γ). (cid:3) ⊆ n Thenextpropositioncompares thesets oflinear coincidenceisometries of shiftsof similar lattices and is the analogue of Lemma 2.5 in [1] for shifted lattices. Proposition 4.8. Let Γ be a lattice in d and x d. R ∈ R (i) If λ + then OC(λx + λΓ) = OC(x + Γ) with Σ (R) = Σ (R) for all λx+λΓ Γ ∈ R R OC(λx+λΓ). ∈ (ii) If S O(d) then OC(Sx+SΓ) =S[OC(x+Γ)]S−1 with Σ (R)= Σ (S−1RS) Sx+SΓ Γ ∈ for all R OC(Sx+SΓ). ∈ Proof. Both statements follow from Theorem 4.3 and Equation (4.2). (cid:3) Now, it is evident from Theorem 4.3 that OC(x + Γ) is a subset of OC(Γ). The set OC(x+Γ) is certainly nonempty because it contains the identity isometry. It also follows from Theorem 4.3 that OC(x + Γ) is closed under inverses, that is, R−1 OC(x+Γ) ∈ whenever R OC(x+Γ). However, given R , R OC(x+Γ), the product R R is not 1 2 2 1 ∈ ∈ necessarily in OC(x+Γ). Thus, one obtains the following result. Proposition 4.9. For a given lattice Γ d and x d, the set OC(x+Γ) is a group if ⊆ R ∈ R and only if it is closed under composition. We shall see in Example 5.25 an instance when OC(x+Γ) fails to form a group. In any case, the productof two linear coincidence isometries of x+Γ whose coincidence indices are relatively prime turns out to be again a linear coincidence isometry of x+Γ. This result is stated in the next proposition. Proposition 4.10. Let Γ d be a lattice and x d. If R ,R OC(x+Γ) with Σ(R ) 1 2 1 ⊆ R ∈ R ∈ and Σ(R ) relatively prime, then R R OC(x+Γ). 2 2 1 ∈ Proof. From Theorem 4.3, R OC(Γ) and R x x Γ+R Γ for j 1,2 . Thus, the j j j ∈ − ∈ ∈ { } product R R OC(Γ). In addition, 2 1 ∈ R R x x = (R R x R x)+(R x x) Γ+R Γ+R R Γ. 2 1 2 1 2 2 2 2 1 − − − ∈ However, R Γ = Γ(R )+R Γ(R ) because Σ(R ) and Σ(R ) are relatively prime (see [40, 2 2 2 1 1 2 Figure 2]). Since Γ(R ) Γ and R Γ(R ) R R Γ, R R x x Γ+R R Γ. The claim 2 2 1 2 1 2 1 2 1 now follows from Theore⊆m 4.3. ⊆ − ∈ (cid:3) THE COINCIDENCE PROBLEM FOR SHIFTED LATTICES AND MULTILATTICES 7 5. Linear coincidences of a shifted square lattice This entire section is devoted to the solution of the coincidence problem for a shifted square lattice. Some of the results here can be found in [26]. 5.1. Solution of the coincidence problem for the square lattice. Let us recall first thecoincidences of thesquarelattice 2 (see[1,28]for details). SinceO(2)is thesemidirect Z product of SO(2) and the cyclic group C generated by the reflection in the x-axis, we 2 restrict our discussion to coincidence rotations at the outset and later on extend it to include coincidence reflections. The group of coincidence rotations of 2 is SOC( 2) = SO(2, ). To determine the Z Z Q structure of this group, the square lattice is identified with the ring of Gaussian integers Γ = [i] = m+ni: m,n ,i2 = 1 embedded in . It can be shown that every coin- Z ∈Z − C cidence rotation in SOC(Γ) by an angle of θ in the counterclockwise direction corresponds (cid:8) (cid:9) to multiplication by the complex number eiθ = εz on the unit circle, where ε 1, i is z ∈ {± ± } a unit in [i] and z is a Gaussian integer with z relatively prime to z. Since the ring [i] Z Z is a Euclidean domain and thus a unique factorization domain, z can be uniquely factored into powers of primes. Hence, a coincidence rotation R of Γ is equivalent to multiplication by the complex number ω np p (5.1) ε , · ω p≡1(4)(cid:18) p(cid:19) Y where n and only a finite number of n = 0, p runs over all rational primes p 1 p p ∈ Z 6 ≡ (mod 4) (called splitting primes in [i]), and ω , and its complex conjugate ω , are the p p Z Gaussian prime factors of p = ω ω . Then z reads p p · (5.2) z = ω np (ω )−np, p p · p≡Y1(4) p≡Y1(4) np>0 np<0 and the coincidence index of R is equal to the numbertheoretic norm of z, Σ(R)= N(z) := z z = z 2. In addition, the CSL obtained from R is the principal ideal Γ(R) = (z) := · | | z [i]. Consequently, the group of coincidence rotations of the square lattice is given by Z SOC( 2) = SO(2, ) = C (ℵ0), where C is the cyclic group of order 4 generated by ∼ 4 4 Z Q ×Z i, and (ℵ0) is the direct sum of countably many infinite cyclic groups each of which is generatZed by ωp. ωp Every coincidence reflection T of 2 can be written as T = R T , where R SOC(Γ) r Z · ∈ and T is the reflection along the real axis (corresponding to complex conjugation). Since r T leaves Γ invariant, Σ(T)= Σ(R) and Γ(T)= Γ(R). Finally, one obtains that OC( 2)= r Z O(2, ) = SOC( 2)⋊ T (where ⋊ stands for semidirect product). r Q Z h i The coincidence indices and the number of CSLs of 2 for a given index m are described Z by means of a generating function. If f (m) denotes the numberof CSLs of 2 of index m, 2 then f is multiplicative (that is, f Z(1) = 1 and f (mn) = f (m)f (nZ) whenever m 2 2 2 2 2 and n Zare relatively prime), and for pZrimes p and r Z , Z Z ∈ N 2, if p 1 (mod 4) f (pr)= ≡ 2 0, otherwise. Z (cid:26) 8 M.J.C.LOQUIASANDP.ZEINER The generating function for f as a Dirichlet series Φ (s) is given by 2 2 Z Z ∞ f (m) 1+p−s 1 ζ (s) 2 (i) (5.3) ΦZ2(s)= Zms = 1 p−s = 1+2−s · Qζ(2s) m=1 p≡1(4) − X Y = 1+ 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 4 + 2 + , 5s 13s 17s 25s 29s 37s 41s 53s 61s 65s 73s ··· where ζ (s) is the Dedekind zeta function of the quadratic field (i) and ζ(s)= ζ (s) is (i) RiemanQn’s zeta function (see [7, 37]). Observefrom (5.3) that the cQoincidence indicesQof the square lattice are positive integers all of whose prime factors are splitting primes in [i]. Z As the rightmost pole of Φ (s) is located at s = 1, we can infer from Delange’s theorem 2 (see for instance, [3, TheoreZm 5 of Appendix]) that the summatory function f (m) m≤N 2 grows asymptotically as N. In other words, the number of CSLs of 2 of index at mZost N π Z P is asymptotically given by N. π The number of coincidence rotations of 2 for a given index m is given by fˆ (m) = 2 4f (m), where the factor 4 stems from thZe fact that 2 has four symmetry rZotations. 2 CoZnsequently, the Dirichlet series generating function forZfˆ is 4Φ (s). 2 2 Z Z Remark 5.1. Observe from the complex number in (5.1) and Equation (5.2) that each coincidence rotation R of Γ = 2 can be associated to a numerator z and unit ε, and this Z shall be written as R . Note however that this correspondence is not unique: one can z,ε take any associate of z as numerator and the unit ε will change accordingly. Nonetheless, throughout this section, R SOC(Γ) stands for multiplication by the complex number z,ε ∈ εz. Furthermore, the fraction z is assumed to bereduced, that is, z and z have no common z z prime factors. Also, we set z = 1 whenever R P(Γ). z,ε ∈ Similarly, T OC(Γ) SOC(Γ) is understood to be the coincidence reflection T = z,ε z,ε ∈ \ R T . z,ε r · 5.2. The sets SOC(x+Γ) and OC(x+Γ). The following lemma gives a criterion when R SOC(Γ) and T OC(Γ) SOC(Γ)are coincidence isometries of the shifted square z,ε z,ε ∈ ∈ \ lattice x+Γ. Lemma 5.2. Let Γ = [i], x , R = R SOC(Γ), and T = R T . z,ε r Z ∈ C ∈ · (i) R SOC(x+Γ) if and only if (εz z)x Γ. ∈ − ∈ (ii) T OC(x+Γ) if and only if εzx zx Γ. ∈ − ∈ Proof. Since z,z are relatively prime and [i] is a principal ideal domain, one has Z Γ+RΓ = Γ+εzΓ = 1gcd(z,z)Γ = 1Γ. z z z ByTheorem4.3,R SOC(x+Γ)ifandonlyifεzx x 1Γ,thatis,whenever(εz z)x Γ. ∈ z − ∈ z − ∈ Because T leaves Γ invariant, Γ +TΓ = 1Γ. Applying again Theorem 4.3 proves the r z corresponding result for T. (cid:3) It turns out that the set of coincidence rotations of x+Γ forms a group. Theorem 5.3. If Γ = [i] then SOC(x+Γ) is a subgroup of SOC(Γ) for all x . Z ∈C Proof. ByProposition4.9,itisenoughtoshowthatSOC(x+Γ)isclosedundercomposition to prove the claim. Let R = R SOC(x+Γ) for j 1,2 . Then (ε z z )x Γ j zj,εj ∈ ∈ { } j j − j ∈ THE COINCIDENCE PROBLEM FOR SHIFTED LATTICES AND MULTILATTICES 9 from Lemma 5.2. Take g := gcd(z ,z ), and express z = h g and z =h g. One has 1 2 1 1 2 2 ε ε h h h h x = 1 ε ε z h h z x 2 1 2 1− 2 1 g 2 1 2 1− 2 1 (cid:0) (cid:1) = 1g(cid:0)ε1h1(ε2z2−z2)x(cid:1)+h2(ε1z1−z1)x ∈ 1gΓ. In the same manner, one obtains that(cid:2) ε ε h h h h x 1Γ. Togeth(cid:3)er, they imply that 2 1 2 1− 2 1 ∈ g ε ε h h h h x(cid:0) 1Γ 1Γ = 1(cid:1)lcm(g,g)Γ = Γ. 2 1 2 1− 2 1 ∈ g ∩ g gg This is because [(cid:0)i] is a principal i(cid:1)deal domain and g is relatively prime to g. Thus, R R = R Z SOC(x+Γ) by Lemma 5.2. (cid:3) 2 1 h2h1,ε2ε1 ∈ However, the situation is more complicated for OC(x+Γ). Onehas the following results. Lemma 5.4. Let Γ = [i] and x . Then OC(x + Γ) is a subgroup of OC(Γ) if Z ∈ C and only if for any coincidence reflections T , T OC(x +Γ), the coincidence rotation 1 2 ∈ T T SOC(x+Γ). 2 1 ∈ Proof. It follows from Proposition 4.9 and Theorem 5.3 that it suffices to show that the productof a coincidence reflection and a coincidence rotation of x+Γ is again a coincidence reflection of x+Γ. Applying the same techniques employed in the proof of Theorem 5.3 yields the claim. (cid:3) Remark 5.5. Letx andT = T OC(x+Γ) SOC(x+Γ)forj 1,2 . Applying ∈ C j zj,εj ∈ \ ∈ { } the procedure used in the proof of Theorem 5.3 to the product T T only leads to 2 1 (5.4) ε ε h h h h x 1Γ, 2 1 2 1− 2 1 ∈ g where g := gcd(z1,z2) and zj = (cid:0)hjg for j 1,2 (cid:1). It follows then from Lemma 5.2 that if ∈ { } z were relatively prime to z , then T T = R (S)OC(x+Γ). This fact can also 1 2 2 1 h2h1,ε2ε1 ∈ be deduced from Proposition 4.10, because if z and z were relatively prime, then so are 1 2 N(z ) = Σ(R ) and N(z )= Σ(R ). 1 1 2 2 Proposition 5.6. Let Γ = [i] and x . If OC(x+Γ) contains a reflection symmetry Z ∈ C T P(Γ) then OC(x+Γ)=SOC(x+Γ)⋊ T and is a subgroup of OC(Γ). Otherwise, the ∈ h i coincidence reflection Tz,ε / OC(x+Γ)forallunitsεof ΓwheneverR = Rz,ε′ SOC(x+Γ) ∈ ∈ for some unit ε′. Proof. Because T P(Γ), T = T for some unitε of [i]. Thus, x εx+Γ by Lemma 5.2. 1,ε ∈ Z ∈ Let T = T OC(x+Γ) SOC(x+Γ) for j 1,2 . If g := gcd(z ,z ) and z = h g j zj,εj ∈ \ ∈ { } 1 2 j j for j 1,2 , then it follows from Lemma 5.2 that ∈ { } g ε ε h h h h x = ε h (ε z x z x) ε h (ε z x z x) Γ. 2 1 2 1 2 1 1 1 2 2 2 1 2 1 1 1 − − − − ∈ Since x εx(cid:0)+ Γ, we have ε(cid:1)ε h h h h x 1Γ. This, together with (5.4), implies ∈ 2 1 2 1 − 2 1 ∈ g that ε ε h h h h x 1 Γ 1Γ = Γ, and thus T T OC(x+Γ). From Lemma 5.4, 2 1 2 1 − 2 1 ∈ g(cid:0) ∩ g (cid:1) 2 1 ∈ OC(x+Γ) is a subgroup of OC(Γ). (cid:0) (cid:1) In addition, any coincidence reflection T′ = Tz′,ε′ of x+Γ can be written as T′ = R′ T · whereR′ = Rz′,εε′ SOC(x+Γ). Hence,OC(x+Γ)isthesemidirectproductofSOC(x+Γ) ∈ and T . h i Suppose OC(x+Γ) does not contain any reflection symmetry and T OC(x+Γ) for z,ε ∈ some unit ε of [i]. Since R SOC(x+Γ), R−1 T OC(x+Γ) by Lemma 5.4. This is z,ε a contradictionZbecause R−1∈T = T P(Γ)·. ∈ (cid:3) · z,ε 1,ε′ε ∈ 10 M.J.C.LOQUIASANDP.ZEINER Proposition 5.6 tells us that when computing for OC(x+Γ), it is advantageous to determine at the outset whether there is a reflection symmetry T that is in OC(x+Γ). If such a T exists, then OC(x+Γ) is a group and it is the semidirect product of SOC(x+Γ) and T . Otherwise, once SOC(x+Γ) has already been identified, only those coincidence h i reflections Tz,ε OC(Γ) for which Rz,ε′ / SOC(x+Γ) for all units ε′ of Γ may be elements ∈ ∈ of OC(x+Γ). The following corollary describes exactly when a reflection symmetry is a coincidence isometry of x+Γ, and thus, gives an explicit version of Proposition 5.6. Corollary 5.7. Let Γ = [i] and x . If one of the following conditions on x is satisfied: Re(x) 1 , Im(x) 1 Z, or Re(x)∈ CIm(x) , then OC(x+Γ) =SOC(x+Γ)⋊ T ∈ 2Z ∈ 2Z ± ∈ Z h 1,εi and is a subgroup of OC(Γ), where 1, if Im(x) 1 ∈ 2Z  1, if Re(x) 1 ε = − ∈ 2Z   i, if Re(x) Im(x)  − ∈ Z i, if Re(x)+Im(x) . − ∈ Z Proof. Apply Lemma 5.2 and Proposition 5.6. (cid:3)  5.3. Determination of SOC(x+Γ) and OC(x+Γ). We now turnto theactual computa- tion of OC(x+Γ) for specific values of x. Given R SOC(Γ), one sees from Lemma 5.2 z,ε ∈ the significance of the expression 2iIm(z), if ε= 1 2Re(z), if ε= 1 (5.5) εz z =  − − −  [Re(z)+Im(z)](1 i), if ε= i − − [Re(z) Im(z)](1+i), if ε= i. − − −   Since εzx zx = ε(zx) (zx), (5.5) can also be used to compute for εzx zx for a given − − − T OC(Γ) SOC(Γ). z,ε ∈ \ Remark 5.8. Let Γ = [i]. Given an R SOC(Γ) or T OC(Γ) SOC(Γ), the z,ε z,ε Z ∈ ∈ \ following holds on account of the choice of z (see (5.2) and Remark 5.1): (i) The rational integers Re(z) and Im(z) are relatively prime and are of different parity (that is, one is odd while the other is even). (ii) Re(z) is nonzero while z = 1 if Im(z) = 0. The following theorem states the complete solution of the coincidence problem for x+Γ whenever x has an irrational component. Theorem 5.9. Let Γ = [i] and x = a+ bi, with a,b . If a or b is irrational then Z ∈ R OC(x+Γ) is a group of at most two elements. In particular, if T , if 2b r (i) a is irrational and b is rational then OC(x+Γ)= h i ∈Z , otherwise. ( {1} T , if 2a 1,−1 (ii) a is rational and b is irrational then OC(x+Γ)= h i ∈ Z , otherwise. ( {1} (iii) both a and b are irrational, and (a) 1, a, and b are rationally independent then OC(x+Γ)= . {1}