Complex Continued Fractions with Constraints on Their Partial Quotients

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1 Complex Continued Fractions with Constraints on Their Partial Quotients Complex Continued Fractions with Constraints on Their Partial Quotients Hans Höngesberg (Wien, Austria) Nicola Oswald (Wuppertal & Würzburg, Germany) Jörn Steuding (Würzburg, Germany) Abstract. It is shown that Hurwitz s continued fraction expansion for complex numbers cannot be applied directly to the ring of integers of a non-quadratic cyclotomic field, however, with a certain modification an analogue of such a continued fraction expansion is derived in the explicit example Q(exp( 2πi )). Moreover, using the geometry of Voronoï diagrams, 8 further generalizations of complex continued fractions are given.. A Brief Account of the History of Complex Continued Fractions Continued fractions of real numbers with applications in and outside mathematics have been studied for millennia. There are several expansions of a given real number into a (convergent) continued fraction possible. The regular continued fraction of a rational number can be computed from the euclidean algorithm for the denominator and the numerator of the reduced fraction; since this algorithm terminates, the continued fraction is finite. For irrational real numbers the expansion into a regular continued fraction is infinite. An alternative expansion is the continued fraction to the nearest integer. Given a real number x [ 2, 2 ), its continued fraction to the nearest integer is of the form ǫ 2 ǫ n x = ǫ a + a a n +..., Key words and phrases: continued fraction, cyclotomic field, lattice, Voronoï diagram 200 Mathematics Subject Classification: A55, J70; 40A5, 52C20

2 2 Hans Höngesberg, Nicola Oswald and Jörn Steuding resp. x = [0,ǫ /a,ǫ 2 /a 2,...,ǫ n /a n,...] for short. The partial quotients a n and signs ǫ n = ± are integers determined by the map x T(x) = x x + for x 0 2 and T(0) = 0 on [ 2, 2 ) by setting ǫ n = ± according to T n (x) being positive or not, and ǫ n a n := T n (x) +, 2 where T k = T T k denotes the kth iteration of T and T 0 is the identity and y standsforthelargestintegerlessthanorequaltoy. Thiscontinuedfraction expansion to the nearest integer was first introduced by Minnigerode [5] (in different notation). Given a continued fraction to the nearest integer, x = [0,ǫ /a,ǫ 2 /a 2,...,ǫ n /a n,...], one can obtain the simple continued fraction by replacing certain partial quotients by relatively simple rules described in 40 of Perron s monography [8] and Dajani et al. [3] in a wider setting. Continued fractions are the method of choice when a rational approximation for a given (irrational) real number is needed. The convergents to a given continued fraction x = [0,ǫ /a,ǫ 2 /a 2,...,ǫ n /a n,...] are defined by the rational numbers x n = [0,ǫ /a,ǫ 2 /a 2,...,ǫ n /a n ]. As Lagrange proved by his law of best approximation, the convergents to a simple continued fraction (as well as the continued fraction to the nearest integer) provide the best possible rational approximations to a given real number. For further details we refer to [8]. The arithmetical theory of continued fractions for complex numbers begins with the work of Adolf Hurwitz [0]. Let S be any set of complex numbers such that i) sum, difference and product of any two elements in S belong to S, ii) any finite domain of the complex plane contains only finitely many points from S (from which already follows that besides zero there is no point from the open unit disk inside S), and, further, iii) S. Starting from some complex number z, Hurwitz built up the following chain of equations: z = a 0 + z, z = a + z 2,..., z n = a n + z n+, wherea n S andnoneofthe z j is assumedto vanish. Thisleadstoacontinued fraction expansion z = a 0 + a + a = [a 0,a,...,a n,z n+ ], + a n + z n+ which one can continue ad infinitum if all z n 0. In modern language, each iteration is determined by the transform T, given by T(0) = 0 and T(z) :=

3 Complex Continued Fractions with Constraints on Their Partial Quotients 3 z [ z] otherwise, where the bracket [z] assigns a certain element from S to z. Supposingfurtherthativ)thenthconvergent pn q n := x n = a 0 + a + a a n (inreducedform)isdistanttoz byaquantitylessthanafixedconstantmultiple of q 2 n, Hurwitz [0] proved that both, the infinite continued fraction z = a 0 + a + a a n +... = [a 0,a,...,a n,...] as well as the sequence of convergents pn q n converge with limit z (which cannot be an element of S); moreover, if z is the solution of a quadratic equation with coefficients from S, then the z n take only finitely many values. With the ring of GaussianintegersZ[i]andtheringofEisensteinintegersZ[ 2 (+ 3)]Hurwitz gave two examples for such a system S; here, as usual, i = denotes the imaginary unit and 2 ( + 3) is a primitive third root of unity, both in the upper half-plane. His elder brother, Julius Hurwitz, investigated in his dissertation [] a related continued fraction expansion with partial quotients from the ideal ( + i)z[i]; see [7] for the interesting historical background. Concerning Hurwitz s assumptions on the system S (that is how he called a set of numbers satisfying conditions i)-iii)), it should be mentioned that S is in fact a ring with the additional assumption that it does not contain any accumulation point. The notion of a ring was introduced by Kronecker and Dedekind in the second half of the nineteenth century; however, rings have been established only in the course of Emmy Noether s conception of modern algebra in the 920s. Dickson [5] was the first to investigate in which quadratic fields Q( D) an analogue of the euclidean algorithm is possible. In the case of imaginary quadratic fields he proved that there exists a euclidean algorithm in the corresponding ring of algebraic integers if, and only if, D =, 2, 3, 7,. His proof for real quadratic fields, however, turned out to be false, and was corrected by Perron [9]. Lunz [3] considered in his dissertation (supervised by Perron) continued fractions in the field Q( 2); already in this case the study of the growth of the denominators of the convergents in absolute value seems to be more difficult than in the Gaussian number field. Similar investigations for several other imaginary quadratic fields are due to Arwin [, 2]. Hilde Gintner proved in her dissertation [8] at the University of Vienna in 936 (supervised by Hofreiter) that in non-euclidean imaginary quadratic number fields one can find examples where the corresponding continued fraction expansion diverges, e.g., z = 2 D if D mod 4, z = 2D+ 2D D if D mod 4. Moreover, she studied diophantine approximation in imaginary quadratic fields not only with continued fractions but using Minkowski s geometry of numbers.

4 4 Hans Höngesberg, Nicola Oswald and Jörn Steuding Summing up: in an imaginary quadratic number field, a continued fraction expansion to the nearest integer is possible if, and only if, the order of the imaginary quadratic field is euclidean. 2. Cyclotomic Fields: Union of Lattices Let n 3 be an integer. Given a primitive n-th root of unity ζ n (e.g., ζ n = exp( 2πi n )), the associated cyclotomic field Q(ζ n) is an algebraic extension of Q of degree ϕ(n), where ϕ(n) is Euler s totient (i.e., the number of prime residue classes modulo n), and its ring of integers is given by Z[ζ n ] (see [6], Chapter, for this and other details about cyclotomic fields). Hurwitz s restriction ii) that his system S shall be discrete (resp., that there shall be only finitely many elements in any finite region of the complex plane) is not valid until n = 3,4,6 (which are exactly the values for which ϕ(n) = 2 and Q(ζ n ) is an imaginary quadratic number field). In fact, for all other values n 3, there exist algebraic integers inside the unit circle: if n 7, then 0 ζ n 2 = 2 2cos 2π n <, giving a contradiction to ii) by taking powers of ζ n ; for n = 5 one finds, by the geometry of the regular pentagon, 0 +ζ = 2 ( 5 ) <. It should be mentioned that Z[ζ 8 ] is norm-euclideanas alreadyshown by Eisenstein [7], vol. II, pp Here the notion norm-euclidean means that the ring in question is euclidean with the canonical norm. Lenstra [2] proved that Z[ζ n ] is norm-euclidean if n 6,24 is a positive integer with ϕ(n) 0. Although Hurwitz s approach does not apply to cyclotomic fields of degree strictly larger than two we shall introduce a modified continued fraction expansion. For the sake of simplicity we consider the explicit example of Q(ζ 8 ) with the primitive eighth root of unity ζ 8 := exp( 2πi 8 ) having degree four over the rationals. Recall that a two-dimensional lattice Ω in C is a discrete additive subgroup. Any such lattice has a representation as Ω = ω Z+ω 2 Z with complex numbers ω and ω 2 being linearly independent over R; this representation is not unique. DefiningafundamentalparallelogrambyF Ω = {0 λ,λ 2 < : λ ω +λ 2 ω 2 }, the set of its translates F Ω (ω) := ω + 2 (ω +ω 2 )+F Ω

5 Complex Continued Fractions with Constraints on Their Partial Quotients 5 with lattice points ω yields a tiling of the complex plane by parallelograms of equal size each of which having exactly one lattice point in the interior which appears to be at its center. We shall call this the lattice tiling of Ω (with respect to the representation Ω = ω Z+ω 2 Z) consisting of lattice parallelograms F Ω (ω). The numbers ζ j 8 with 0 j < 4 = ϕ(8) form an integral basis for Z[ζ 8]; obviously,wemayalsochoose{,i,ζ 8,ζ 8 }asintegralbasis. NotethatQ(ζ 8 +ζ 8 ) is the maximal real subfield of Q(ζ 8 ). We shall associate two lattices. The first lattice is given by Λ := Z+Zi ( = Z[i]). For a complex number z we have z F Λ (a + ib) with some lattice point a + ib Λ by construction, and we write [z] = a + ib for the lattice point associated with z in this way. Notice that [z] is the closest lattice point to z, however, for general lattices this is not true. In fact, for any element from a parallelogram F Ω (ω) the interior lattice point is the nearest lattice point (in euclidean distance) if, and only if, the diagonals of the parallelogram are of equal length, i.e., F Ω (ω) is rectangular. This holds true for Λ as well as for the second lattice we shall consider, namely the one defined by Λ 2 := Zζ 8 +Zζ 8. Here we shall write [z] 2 = cζ 8 + dζ 8 for the lattice point cζ 8 + dζ 8 such that z F Λ2 (cζ 8 +dζ 8 ). Notice that also Λ 2 is rectangular; actually both, Λ and Λ 2 are even quadratic as follows from the geometry of the eighth roots of unity. In order to have a unique assignment on the boundary of our lattices we may assume that in such cases the larger coefficient shall be chosen. Finally, let (2.) [z] := 2 ([z] +[z] 2 ) = 2 (a+bi+cζ 8 +dζ 8 ) =: (a,b,c,d),2 denote the arithmetical mean of the associated lattice points. It follows that [z] is half an algebraic integer, i.e., an element of 2 Z[ζ 8]. The union of the lattices, Λ Λ 2, is again a discrete set of complex numbers but is neither a lattice nor a system S in the sense of Hurwitz [0]. The lattice tilings of Λ and Λ 2 provide a tiling of the complex plane in polygons by subdividing the parallelograms of the respective lattices into smaller polygons which we shall denote by Z((a,b,c,d),2 ) according to the unique assignment of the half algebraic integer (a,b,c,d),2 = 2 (a+bi+cζ 8 +dζ 8 ) (see Figure below). Following Hurwitz we consider the sequence of equations (2.2) z = a 0 + z, z = a + z 2,..., z n = a n + z n+ with a n = [z n ] = (a,b,c,d),2 ; here the z n are assumed not to vanish. This

6 6 Hans Höngesberg, Nicola Oswald and Jörn Steuding leads to a continued fraction expansion (2.3) z = a 0 + a + a = [a 0,a,...,a n,z n+ ] + a n + z n+ having partial quotients in the set 2 Z[ζ 8]. Similarly to Hurwitz s continued fraction this expansion can be described by z T(z) = z [ z ], where the Gauß bracket is replaced by [ ] defined in (2.). Obviously, a vanishing z n would imply a finite expansion going along with z Q(ζ 8 ). In the sequel we shall assume z Q(ζ 8 ) in order to have an infinite continued fraction. Im 2 (2,0,,),2 2 P 2 Re ( 2,, 2, ),2 R 2 Figure. The union of the lattices Λ and Λ 2 As in the case of Hurwitz s complex continued fraction certain sequences of partial quotients are impossible. By construction, z n a n lies inside an

7 Complex Continued Fractions with Constraints on Their Partial Quotients 7 icosikaitetragon or, in simpler words, a 24 sided polygon which we denote as P with center at the origin, and is determined by the straight lines x = ± 2, y = ± 2, x = ±( ), y = ±( ), y ± x = ±( ) and x±y = ± 2 2 defining the boundary in the x+iy-plane (see Figure ). Hence, z n+ = z n a n R := P ; here we have used the notation M := {m : mp M} for any set M not containing zero. In the sequel we shall also use the notation D r (m) (resp. D r (m)) for the open (closed) disk of radius r with center m. Therefore, the following half algebraic integers cannot occur as partial quotients: (0,0,0,0),2, (±,0,0,0),2,(0,±,0,0),2,(0,0,±,0),2,(0,0,0,±),2, (±,0,0,±),2,(±,0,±,0),2,(0,±,±,0),2,(0,±,0, ),2, (±,0,±,±),2,(±,±,±,0),2,(0,±,±, ),2,(±,,±0,±),2, (±,±,±,±),2,(±,±,±, ),2,(,±,±, ),2,(±,,±,±),2, (±2,±,±,±),2,(±,±,±2,±),2,(±,±,±2, ),2,(±,±2,±, ),2, (,±2,±, ),2,(,±,±, 2),2,(,±,, 2),2,( 2,±,, ),2. Next we investigate the sequence of partial quotients a n with respect to convergence. Suppose that a n = (2,0,,),2, then z n+ Z((2,0,,),2 ). In view of (2.2) we have z n a n Z ((2,0,,),2 ). The latter set is bounded by D2 6 ( i), D2 6 ( i), x±y = 2 2, and x = Hence, the set Z ((2,0,,),2 ) intersects with the real axis at x = 3 2. However, the polygons Z(( 2,0,, ),2 ), Z(( 2,,0, 2),2 ), Z(( 2,, 2,0),2 ), Z((,,0, 2),2 ), and Z((,, 2,0),2 ) have all in common that their respective lattice points have distance at least 3 2 in x-direction to the boundary. A similar reasoning provides restrictions for their predecessors of a n = ( 2,, 2, ),2. This leads to a list of pairs which do not occur as consecutive partial quotients (see the table on the next page). In order to prove the convergence of this continued fraction expansion we shall show (2.4) k n > with k n := q n q n by induction on n. This implies convergence since by the standard machinery of continued fraction calculus one has z p n ( ) n = q n qn(z 2 n+ +kn ) and z p n q n = ( ) n qn 2 (z n+ +k n).

8 8 Hans Höngesberg, Nicola Oswald and Jörn Steuding a n a n+ ( 2,0,, ),2, (2,0,,),2 ( 2,,0, 2),2, ( 2,, 2,0),2, (,,0, 2),2, (,, 2,0),2 (2,0,,),2, (2,,0,2),2, ( 2,, 2, ),2 (,,0,2),2 Table. Impossible pairs of consecutive partial quotients Here p j and q j denote the numerator and denominator to the convergents of the continued fraction expansion defined in the same way as in the previous section. Moreover, we shall use the recursive formula k n = a n + k n. For k = a assertion (2.4) obviously holds since a n R. Now assume k j > for j < n and k n with some positive integer n. Since k n = a n + k n D (a n ) and, by assumption, k n, it follows that a n has to be one of the following numbers: (±2,0,±,±),2,(±,±,±2,0),2,(0,±2,±, ),2, (,±,0, 2),2. By symmetry, we may assume without loss of generality that a n = (2,0,,),2 = Hence, k n = a n + k n is located in the intersection of the unit disk and D ((2,0,,),2 ). Consequently, k n = k n a n lies in the intersection of the unit disk and D (( 2,0,, ),2 ). Hence, k n = = a n + k n a n k n 2 is located outside the unit disk but in the interior of D 7 ( 2+4 2) ( 7 ( 2 3 2)) (the set in Figure coloured in green). Since k n 2 > it follows that k n lies as well in D (a n ). Hence, a n can take only one of the following values: ( 2,0,, ),2,( 2,, 2,0),2,( 2,,0, 2),2,(,,0, 2),2, (,, 2,0),2,( 2,, 2, ),2,( 2,,, 2),2. In view ofour list of impossible partial quotients (see the table above) the value for a n can be found amongst ( 2,, 2, ),2,( 2,,, 2),2.

9 Complex Continued Fractions with Constraints on Their Partial Quotients 9 Again, by symmetry, we may suppose without loss of generality that a n = ( 2,, 2, ),2. It follows that k n = a n + k n 2 lies in the intersection of the disks D 7 ( 2+4 2) ( 7 ( 2 3 2)) and D (( 2,, 2, ),2 ). Hence, k n 2 = k n a n is in the intersection of the unit disk and D 7 ( 2+4 2) ( 7 ( )+ 2 (+ 2 2)i) D0.53 ( i). Thus, we find k n 2 outside the unit disk and inside D 0.3 ( i) (the set in Figure above coloured in brown). Since k n 2 = a n 2 + k n 3 lies inside D (b n 2 ), we conclude that a n 2 has to be one of the following numbers: (,,0,2),2,(2,,0,2),2,(2,0,,),2. However, all these values appear in the list of impossible partial quotients (see the table on the previous page), giving the desired contradiction. Thus we have proved Theorem. The continued fraction expansion (2.3) with partial quotients (2.) from 2 Z[ζ 8] converges. To overcome the minor flaw that the partial quotients might be not algebraic integersonemayexchangeλ andλ 2 bytakingtheirsublattices2λ = 2Z+2iZ and 2Λ 2 = 2ζ 8 Z + 2ζ 8 Z and follow the above analysis of the corresponding continued fraction expansion. There are several aspects which could be studied. Firstly, what are the arithmetical properties of this new continued fraction expansion? Can one prove a similar result on bounded expansions and quadratic equations as Hurwitz did for his complex continued fractions? Moreover, what are the limits of the construction for Q(ζ 8 ) sketched above? Does this lead to continued fraction expansions for other cyclotomic fields as well? We do not answer these questions here but provide another generalization of Hurwitz s approach to complex continued fractions. 3. Using Voronoï Diagrams for Continued Fraction Expansions There is a lot of literature about Voronoï diagrams and Voronoï cells; the monographies of Gruber [9] and Matousek [4] provide excellent readings on this topic. In the sequel we shall concentrate on the two-dimensional situation. Given a discrete set S of points in the complex plane, the Voronoï cell for a point p S is defined by V S (p) = {z C : z p z q q S},

10 0 Hans Höngesberg, Nicola Oswald and Jörn Steuding i.e., the set of all z that are closer to p than to any other element of S (in euclidean norm). Any Voronoï cell V S (p) is a convex polygon and their union over all p S is called Voronoï diagram and yields a tiling of the complex plane. The earliest appearance of Voronoï cells is in a picture in Descartes solar system in his Principia Philosophiae from 644 (cf. [4], p. 20). A rigour mathematical definition was first given by Dirichlet [6] and Voronoï [20] in the setting of quadratic forms. The Voronoï diagram of the lattice Z[i] of Gaussian integers coincides with the lattice tiling by squares F Z[i] (a + ib) introduced in the previous section. We have already noticed there, although in different language, that this is a rare event, namely, that a lattice tiling is a Voronoï diagram if, and only if, the lattice is rectangular. Otherwise the Voronoï cells are hexagonal (see also [9]). Figure 2. On the left a random Voronoï diagram. On the right the one for the lattice generated by and 4 ( + 3i); here the cells are pretty similar to honeycombs. In the sequel we shall consider lattices of the form Λ = δz+τz with a real number δ > 0 and τ = x + iy C from the upper half-plane (i.e., y > 0). This is not a severe restriction since we are concerned with approximations by fractions p q built from our lattice, p,q Λ, and (3.) with P = ω δ ω p,q = δ q Ω, where ω p q = ω δ q = P Q δ p Ω := ω Z+ω 2 Z = ω δ (δz+τz) = ω δ Λ by setting τ = δ ω2 ω (which is not real by the linear independence of ω and ω 2 over R and, hence, can be chosen as an element from the upper half-plane).

11 Complex Continued Fractions with Constraints on Their Partial Quotients Therefore, any approximation by a quotient from Λ corresponds to an approximation by a quotient of the equivalent lattice Ω and vice versa. Lattices Ω and Ω 2 are said to be equivalent if there exists a complex number ω 0 such that Ω = ωω 2. Similarly to Hurwitz and (2.2) and (2.3), respectively, we consider a sequence of equations, z = a 0 + z, z = a + z 2,..., z n = a n + z n+ with a n Λ chosen such that z n is in the Voronoï cell V Λ (a n ) around a n ; of course, the appearing z j are assumed not to vanish. This yields to a continued fraction expansion (3.2) z = a 0 + a + a = [a 0,a,...,a n,z n+ ] + a n + z n+ with partial quotients in the lattice Λ. Obviously, a vanishing z n would imply that z has a finite expansion and, thus, z would have a representation as a quotient of two lattice points. In the sequel we shall assume that z is not of this type, that is z C \ Q(Λ), where Q(Λ) := { p q : p,q Λ}, and the continued fraction expansion for z is infinite. In order to prove the convergence of this continued fraction expansion we define once again k n = qn q n and show the analogue of (2.4), i.e., k n > ; here q n denotes the denominators of the nth convergent to the just defined new continued fraction for z. It is not too difficult to deduce the desired convergence in just the same way as (2.4) implied convergence of the continued fraction expansion considered in the previous section. In this general setting our reasoning shall be less precise than in the explicit example of the previous section. By definition, we find for the Voronoï cell V Λ (0) D ρ (0) with (3.3) ρ := 2 max{δ, τ, τ ±δ }; this follows by considering the neighbouring lattice points ±δ,±τ,±τ±δ of the origin. Of course, here one could be more precise by exploiting the geometry and using the knowledge that the volume of each cell equals the determinant of the lattice. Since a n z n V Λ (0) it follows that z n+ D ρ (0) = C\D ρ (0). Hence, z n+ lies outside the disk of radius ρ with center at the origin. In order to prevent that z n+ is located inside the Voronoï cell V Λ (0) (which would cause difficulties for convergence) we need to put a restriction on ρ. In

12 2 Hans Höngesberg, Nicola Oswald and Jörn Steuding view of z n+ V Λ (a n+ ) and a n+ z n+ ρ we obtain a n+ ρ ρ. To conclude with the proof by induction we assume k n > and deduce via k n+ = a n+ + k n the inequality k n+ = a n+ k n > ρ ρ which is greater than or equal to one for ρ 2. Hence, Theorem. The continued fraction expansion (3.2) with partial quotients in the lattice Λ = δz + τz converges provided ρ 2, where ρ is given by (3.3). The bound on ρ is not completely satisfying. Indeed, the statement of the theorem does not imply the cases of the Gaussian lattice Z[i] and the Eisenstein latticez[ 2 (+ 3)]consideredbyHurwitz[0]. Amoresophisticatedanalysis should lead to an extension of the above theorem covering these cases. Another, more simple solution, relies on the observation (3.) that for approximation by quotients from a lattice one may exchange the lattice in question by an equivalent lattice. Hence, by using an appropriate scaling, one can obtain a continued fraction expansion with partial quotients from any given lattice. Figure 3. The restrictions for ρ are indicated by the circle in the middle ( τ 2( 2 )) and the neighbouring circles ( τ ±δ 2( 2 )) with the special value δ = 2. The set of admissible τ is given by the non-empty intersection of all three circles. Inasimilarwayonecouldalsoconsiderarbitrarydiscretepoint setss in the complex plane in place of a lattice provided the corresponding Voronoï diagram would share the essential property of having sufficiently small Voronoï cells. This would lead to another continued fraction expansion with partial quotients

13 Complex Continued Fractions with Constraints on Their Partial Quotients 3 from S, however, for practical purposes discrete sets S with structure seem to be more useful than others. Acknowledgments. The second and third author would like to express their gratitude to Dr. Halyna Syta for the organization of the Fifth International Conference on Analytic Number Theory and Spatial Tesselations at the National Pedagogical Dragomanov University Kiev in September 203 in honour of Georgii Voronoï and her kind hospitality. References [] A. Arwin, Einige periodische Kettenbruchentwicklungen, J. f. M. 55 (926), -28 [2] A. Arwin, Weitere periodische Kettenbruchentwicklungen, J. f. M. 59 (928), [3] K. Dajani, D. Hensley, C. Kraaikamp, V. Masarotto, Arithmetic and ergodic properties of flipped continued fraction algorithms, Acta Arith. 53 (202), 5-79 [4] R. Descartes, Principia Philosophiae, 644 [5] L.E. Dickson, Algebren und ihre Zahlentheorie, Zürich & Leipzig 927 [6] P.G.L. Dirichlet, Über die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen, J. Reine Angew. Math. 40 (850), [7] G. Eisenstein, Mathematische Werke, Chelsea, New York 975 [8] H. Gintner, Ueber Kettenbruchentwicklung und über die Approximation von komplexen Zahlen, Dissertation, University of Vienna, 936 [9] P.M. Gruber, Convex and Discrete Geometry, Springer 2007 [0] A. Hurwitz, Ueber die Entwickelung complexer Grössen in Kettenbrüche, Acta Math. XI (888), [] J. Hurwitz, Ueber eine besondere Art der Kettenbruch-Entwickelung complexer Grössen, Dissertation at the University of Halle, 895 [2] H.W. jun. Lenstra, Euclid s algorithm in cyclotomic fields, J. Lond. Math. Soc. 0 (975), [3] P. Lunz, Kettenbrüche, deren Teilnenner dem Ring der Zahlen und 2 angehören, Diss. München, A. Ebner München 937 [4] J. Matoušek, Lectures on Discrete Geometry, Springer 2002 [5] C. Minnigerode, Ueber eine neue Methode, die Pell sche Gleichung aufzulösen, Gött. Nachr. (873), [6] J. Neukirch, Algebraic number theory, Springer 992 [7] N. Oswald, J. Steuding, Complex Continued Fractions Early Work of the Brothers Adolf and Julius Hurwitz, Arch. Hist. Exact Sci. 68, No. 4 (204),

14 4 Hans Höngesberg, Nicola Oswald and Jörn Steuding [8] O. Perron, Die Lehre von den Kettenbrüchen, Teubner, Leipzig, st ed. 93; 2nd ed. 929; 3rd ed. 954 in two volumes [9] O. Perron, Quadratische Zahlkörper mit Euklidischem Algorithmus, Math. Ann. 07 (932), [20] G.F. Voronoï, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxieme mémoire: recherches sur les paralléloèdres primitifs, J. Reine Angew. Math. 34 (908), ; 36 (909), Hans Höngesberg Gunoldstraße 4/2, 90 Vienna, Austria Nicola Oswald Department of Mathematics and Informatics, University of Wuppertal Gaußstr. 20, 42 9 Wuppertal, Germany oswald@uni-wuppertal.de and Department of Mathematics, Würzburg University Emil Fischer-Str. 40, Würzburg, Germany nicola.oswald@mathematik.uni-wuerzburg.de Jörn Steuding Department of Mathematics, Würzburg University Emil Fischer-Str. 40, Würzburg, Germany steuding@mathematik.uni-wuerzburg.de