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Complex Continued Fractions with Constraints on Their Partial Quotients
1
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 Voronoı̈ diagrams,
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further generalizations of complex continued fractions are given.
1.
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 ∈ [− 21 , 12 ), its continued fraction to the nearest integer is of the form
x=
ǫ1 ǫ2
ǫn
...
...,
a1 + a2 +
+ an +
Key words and phrases: continued fraction, cyclotomic field, lattice, Voronoı̈ diagram
2010 Mathematics Subject Classification: 11A55, 11J70; 40A15, 52C20
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Hans Höngesberg, Nicola Oswald and Jörn Steuding
resp. x = [0, ǫ1 /a1 , ǫ2 /a2 , . . . , ǫn /an , . . .] for short. The partial quotients an
and signs ǫn = ±1 are integers determined by the map
1
1
1
x 7→ T (x) =
for x 6= 0
−
+
|x|
|x| 2
and T (0) = 0 on [− 21 , 12 ) by setting ǫn = ±1 according to T n−1 (x) being
positive or not, and
1
ǫn
,
an :=
+
T n−1 (x) 2
where T k = T ◦ T k−1 denotes the kth iteration of T and T 0 is the identity and
⌊y⌋ stands for the largest integer less than or equal to y. This continued fraction
expansion to the nearest integer was first introduced by Minnigerode [15] (in
different notation). Given a continued fraction to the nearest integer, x =
[0, ǫ1 /a1 , ǫ2 /a2 , . . . , ǫn /an , . . .], one can obtain the simple continued fraction by
replacing certain partial quotients by relatively simple rules described in §40 of
Perron’s monography [18] 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, ǫ1 /a1 , ǫ2 /a2 , . . . , ǫn /an , . . .] are defined by the rational
numbers xn = [0, ǫ1 /a1 , ǫ2 /a2 , . . . , ǫn /an ]. 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 [18].
The arithmetical theory of continued fractions for complex numbers begins
with the work of Adolf Hurwitz [10]. 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) 1 ∈ S. Starting from some complex
number z, Hurwitz built up the following chain of equations:
z = a0 +
1
,
z1
z 1 = a1 +
1
,
z2
... ,
z n = an +
1
zn+1
,
where an ∈ S and none of the zj is assumed to vanish. This leads to a continued
fraction expansion
z = a0 +
1
1
1
1
= [a0 , a1 , . . . , an , zn+1 ],
...
a1 + a2 +
+ an + zn+1
which one can continue ad infinitum if all zn 6= 0. In modern language, each
iteration is determined by the transform T , given by T (0) = 0 and T (z) :=
Complex Continued Fractions with Constraints on Their Partial Quotients
3
− z1 otherwise, where the bracket [z] assigns a certain element from S to z.
Supposing further that iv) the nth convergent pqnn := xn = a0 + a11 + a12 + . . . + a1n
(in reduced form) is distant to z by a quantity less than a fixed constant multiple
of q12 , Hurwitz [10] proved that both, the infinite continued fraction
1
z
n
z = a0 +
1
1
1
...
. . . = [a0 , a1 , . . . , an , . . .]
a1 + a2 +
+ an +
as well as the sequence of convergents pqnn 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 zn take only finitely many values. With
√ the ring of
Gaussian integers Z[i] and the ring of Eisenstein integers Z[ 12 (1+ −3)] Hurwitz
√
gave two examples for such√a system S; here, as usual, i = −1 denotes the
imaginary unit and 21 (1 + −3) is a primitive third root of unity, both in
the upper half-plane. His elder brother, Julius Hurwitz, investigated in his
dissertation [11] a related continued fraction expansion with partial quotients
from the ideal (1 + i)Z[i]; see [17] 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 1920s.
√
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 = −1, −2, −3, −7, −11.
His proof for real quadratic fields, however, turned out to be false, and was corrected by Perron [19]. Lunz [13] 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 [1, 2]. Hilde Gintner
proved in her dissertation [8] at the University of Vienna in 1936 (supervised
by Hofreiter) that in non-euclidean imaginary quadratic number fields one can
find examples where the corresponding continued fraction expansion diverges,
e.g.,
√
√
D if D ≡ 1 mod 4.
z = 21 D if D 6≡ 1 mod 4, z = 2D+1
2D
Moreover, she studied diophantine approximation in imaginary quadratic fields
not only with continued fractions but using Minkowski’s geometry of numbers.
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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 ζn (e.g., ζn =
exp( 2πi
n )), the associated cyclotomic field Q(ζn ) 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[ζn ] (see [16], Chapter
1, 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(ζn ) 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 6= |1 − ζn |2 = 2 − 2 cos 2π
n < 1,
giving a contradiction to ii) by taking powers of 1 − ζn ; for n = 5 one finds, by
the geometry of the regular pentagon,
√
0 6= |1 + ζ53 |2 = 21 ( 5 − 1) < 1.
It should be mentioned that Z[ζ8 ] is norm-euclidean as already shown by Eisenstein [7], vol. II, pp. 585-595. Here the notion ’norm-euclidean’ means that
the ring in question is euclidean with the canonical norm. Lenstra [12] proved
that Z[ζn ] is norm-euclidean if n 6= 16, 24 is a positive integer with ϕ(n) ≤ 10.
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(ζ8 ) with
the primitive eighth root of unity ζ8 := 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 = ω1 Z + ω2 Z with complex numbers
ω1 and ω2 being linearly independent over R; this representation is not unique.
Defining a fundamental parallelogram by FΩ = {0 ≤ λ1 , λ2 < 1 : λ1 ω1 +λ2 ω2 },
the set of its translates
FΩ (ω) := ω + 21 (ω1 + ω2 ) + FΩ
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 = ω1 Z + ω2 Z) consisting of lattice parallelograms
FΩ (ω).
The numbers ζ8j with 0 ≤ j < 4 = ϕ(8) form an integral basis for Z[ζ8 ];
obviously, we may also choose {1, i, ζ8 , ζ8 } as integral basis. Note that Q(ζ8 +ζ8 )
is the maximal real subfield of Q(ζ8 ). We shall associate two lattices. The first
lattice is given by
Λ1 := Z + Zi ( = Z[i]).
For a complex number z we have z ∈ FΛ1 (a + ib) with some lattice point
a + ib ∈ Λ1 by construction, and we write [z]1 = a + ib for the lattice point
associated with z in this way. Notice that [z]1 is the closest lattice point to
z, however, for general lattices this is not true. In fact, for any element from
a parallelogram FΩ (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., FΩ (ω) is rectangular. This holds true for Λ1 as well as for
the second lattice we shall consider, namely the one defined by
Λ2 := Zζ8 + Zζ8 .
Here we shall write [z]2 = cζ8 + dζ8 for the lattice point cζ8 + dζ8 such that
z ∈ FΛ2 (cζ8 + dζ8 ). Notice that also Λ2 is rectangular; actually both, Λ1 and
Λ2 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
(2.1)
[z] := 21 ([z]1 + [z]2 ) = 12 (a + bi + cζ8 + dζ8 ) =: (a, b, c, d)1,2
denote the arithmetical mean of the associated lattice points. It follows that
[z] is half an algebraic integer, i.e., an element of 21 Z[ζ8 ]. The union of the
lattices, Λ1 ∪ Λ2 , is again a discrete set of complex numbers but is neither
a lattice nor a system S in the sense of Hurwitz [10]. The lattice tilings of
Λ1 and Λ2 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)1,2 ) according to the unique assignment of the half
algebraic integer (a, b, c, d)1,2 = 12 (a + bi + cζ8 + dζ8 ) (see Figure 1 below).
Following Hurwitz we consider the sequence of equations
(2.2)
z = a0 +
1
,
z1
z 1 = a1 +
1
,
z2
... ,
z n = an +
1
zn+1
with an = [zn ] = (a, b, c, d)1,2 ; here the zn are assumed not to vanish. This
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Hans Höngesberg, Nicola Oswald and Jörn Steuding
leads to a continued fraction expansion
(2.3)
z = a0 +
1
1
1
1
= [a0 , a1 , . . . , an , zn+1 ]
...
a1 + a2 +
+ an + zn+1
having partial quotients in the set 12 Z[ζ8 ]. Similarly to Hurwitz’s continued
fraction this expansion can be described by z 7→ T (z) = z1 − [ z1 ], where the
Gauß bracket ⌊ · ⌋ is replaced by [ · ] defined in (2.1). Obviously, a vanishing zn
would imply a finite expansion going along with z ∈ Q(ζ8 ). In the sequel we
shall assume z 6∈ Q(ζ8 ) in order to have an infinite continued fraction.
Im
b
b
b
b
2
b
b
b
b
b
b
b
b
1
b
b
b
b
b
b
b
b
b
(2, 0, 1, 1)1,2
b
b
−2
b
b
−1
b
1
P
b
b
b
b
b
Re
2
b
b
b
−1
b
b
(−2, −1, −2, −1)1,2
b
b
b
b
b
b
R
b
b
b
−2
b
b
b
b
b
b
Figure 1. The union of the lattices Λ1 and Λ2
As in the case of Hurwitz’s complex continued fraction certain sequences
of partial quotients are impossible. By construction, zn − an lies inside an
Complex Continued Fractions with Constraints on Their Partial Quotients
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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 = ± 12 ,
√
√
√
y = ± 21 , x = ±( 41 + 14 2), y = ±( 41 + 14 2), y ± x = ±( 12 + 14 2) and
√
x ± y = ± 12 2 defining the boundary in the x + iy-plane (see Figure 1). Hence,
zn+1 =
1
∈ R := P−1 ;
z n − an
here we have used the notation M−1 := {m−1 : mp ∈ M} for any set M
not containing zero. In the sequel we shall also use the notation Dr (m) (resp.
Dr (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)1,2,
(±1, 0, 0, 0)1,2, (0, ±1, 0, 0)1,2, (0, 0, ±1, 0)1,2, (0, 0, 0, ±1)1,2,
(±1, 0, 0, ±1)1,2, (±1, 0, ±1, 0)1,2, (0, ±1, ±1, 0)1,2, (0, ±1, 0, ∓1)1,2,
(±1, 0, ±1, ±1)1,2, (±1, ±1, ±1, 0)1,2, (0, ±1, ±1, ∓1)1,2, (±1, ∓1, ±0, ±1)1,2,
(±1, ±1, ±1, ±1)1,2, (±1, ±1, ±1, ∓1)1,2, (∓1, ±1, ±1, ∓1)1,2, (±1, ∓1, ±1, ±1)1,2,
(±2, ±1, ±1, ±1)1,2, (±1, ±1, ±2, ±1)1,2, (±1, ±1, ±2, ∓1)1,2, (±1, ±2, ±1, ∓1)1,2,
(∓1, ±2, ±1, ∓1)1,2, (∓1, ±1, ±1, ∓2)1,2, (∓1, ±1, ∓1, ∓2)1,2, (∓2, ±1, ∓1, ∓1)1,2.
Next we investigate the sequence of partial quotients an with respect to
convergence. Suppose that an = (2, 0, 1, 1)1,2, then zn+1 ∈ Z((2, 0, 1, 1)1,2 ). In
−1
view of
we have zn −
). The latter set is bounded
by
√ (2.2) √
√an ∈ Z√ ((2, 0, 1, 1)1,2√
√
D 26 ( 16 2 + 16 2i), D 62 ( 16 2 − 61 2i), x ± y = 21 2, and x = 41 + 41 2. Hence,
√
the set Z−1 ((2, 0, 1, 1)1,2 ) intersects with the real axis at x = 31 2. However,
the polygons Z((−2, 0, −1, −1)1,2), Z((−2, 1, 0, −2)1,2), Z((−2, −1, −2, 0)1,2),
Z((−1, 1, 0, −2)1,2), and Z((−1, −1, −2, 0)1,2) have √
all in common that their
respective lattice points have distance at least 13 2 in x-direction to the
boundary. A similar reasoning provides restrictions for their predecessors of
an = (−2, −1, −2, −1)1,2. 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
qn
(2.4)
|kn | > 1
with kn :=
qn−1
by induction on n. This implies convergence since by the standard machinery
of continued fraction calculus one has
z−
(−1)n
pn
= 2
qn
qn (zn+1 + kn−1 )
and
z−
(−1)n−1
pn−1
= 2
.
−1
qn−1
qn−1 (zn+1
+ kn )
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Hans Höngesberg, Nicola Oswald and Jörn Steuding
an
(−2, 0, −1, −1)1,2,
(−2, 1, 0, −2)1,2,
(−2, −1, −2, 0)1,2,
(−1, 1, 0, −2)1,2,
(−1, −1, −2, 0)1,2
(2, 0, 1, 1)1,2,
(2, −1, 0, 2)1,2,
(1, −1, 0, 2)1,2
an+1
(2, 0, 1, 1)1,2
(−2, −1, −2, −1)1,2
Table 1. Impossible pairs of consecutive partial quotients
Here pj and qj denote the numerator and denominator to the convergents of
the continued fraction expansion defined in the same way as in the previous
1
section. Moreover, we shall use the recursive formula kn = an + kn−1
.
For k1 = a1 assertion (2.4) obviously holds since an ∈ R. Now assume
|kj | > 1 for 1 ≤ j < n and |kn | ≤ 1 with some positive integer n. Since
1
kn = an + kn−1
∈ D1 (an ) and, by assumption, |kn | ≤ 1, it follows that an has
to be one of the following numbers:
(±2, 0, ±1, ±1)1,2, (±1, ±1, ±2, 0)1,2, (0, ±2, ±1, ∓1)1,2,
(∓1, ±1, 0, ∓2)1,2.
By symmetry, we may assume without loss of generality that
√
an = (2, 0, 1, 1)1,2 = 1 + 21 2.
1
Hence, kn = an + kn−1
is located in the intersection of the unit disk and
1
= kn − an lies in the intersection of the
D1 ((2, 0, 1, 1)1,2 ). Consequently, kn−1
unit disk and D1 ((−2, 0, −1, −1)1,2). Hence,
kn−1 =
1
1
= an−1 +
kn − an
kn−2
√
is located outside the unit disk but in the interior of D 1 (−2+4√2) ( 17 (−2 − 3 2))
7
(the set in Figure 1 coloured in green). Since |kn−2 | > 1 it follows that kn−1
lies as well in D1 (an−1 ). Hence, an−1 can take only one of the following values:
(−2, 0, −1, −1)1,2, (−2, −1, −2, 0)1,2, (−2, 1, 0, −2)1,2, (−1, 1, 0, −2)1,2,
(−1, −1, −2, 0)1,2, (−2, −1, −2, −1)1,2, (−2, 1, −1, −2)1,2.
In view of our list of impossible partial quotients (see the table above) the value
for an−1 can be found amongst
(−2, −1, −2, −1)1,2, (−2, 1, −1, −2)1,2.
Complex Continued Fractions with Constraints on Their Partial Quotients
9
Again, by symmetry, we may suppose without loss of generality that an−1 =
1
(−2, −1, −2, −1)1,2. It follows that kn−1 = an−1 + kn−2
lies in the intersection
√
1
√
of the disks D 1 (−2+4 2) ( 7 (−2 − 3 2)) and D1 ((−2, −1, −2, −1)1,2). Hence,
7
1
kn−2 = kn−1 − an−1 is in the intersection of the unit disk and
√
√
D 1 (−2+4√2) ( 17 (5 + 49 2) + 12 (1 + 12 2)i) ⊆ D0.53 (1.17 + 0.85i).
7
Thus, we find kn−2 outside the unit disk and inside D0.3 (0.65 + 0.47i) (the set
1
in Figure 1 above coloured in brown). Since kn−2 = an−2 + kn−3
lies inside
D1 (bn−2 ), we conclude that an−2 has to be one of the following numbers:
(1, −1, 0, 2)1,2, (2, −1, 0, 2)1,2, (2, 0, 1, 1)1,2 .
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 (2.1)
from 21 Z[ζ8 ] converges.
To overcome the minor flaw that the partial quotients might be not algebraic
integers one may exchange Λ1 and Λ2 by taking their sublattices 2Λ1 = 2Z+2iZ
and 2Λ2 = 2ζ8 Z + 2ζ8 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(ζ8 ) 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 Voronoı̈ Diagrams for Continued Fraction Expansions
There is a lot of literature about Voronoı̈ diagrams and Voronoı̈ cells; the
monographies of Gruber [9] and Matousek [14] 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 Voronoı̈ cell for
a point p ∈ S is defined by
VS (p) = {z ∈ C : |z − p| ≤ |z − q| ∀q ∈ S},
10
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 Voronoı̈ cell VS (p) is a convex polygon and their union
over all p ∈ S is called Voronoı̈ diagram and yields a tiling of the complex
plane. The earliest appearance of Voronoı̈ cells is in a picture in Descartes’
solar system in his Principia Philosophiae from 1644 (cf. [14], p. 120). A
rigour mathematical definition was first given by Dirichlet [6] and Voronoı̈ [20]
in the setting of quadratic forms.
The Voronoı̈ diagram of the lattice Z[i] of Gaussian integers coincides with
the lattice tiling by squares FZ[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 Voronoı̈ diagram if, and only if, the
lattice is rectangular. Otherwise the Voronoı̈ cells are hexagonal (see also [9]).
Figure 2. On the left a random Voronoı̈ diagram. On the right the one for
the lattice generated by 1 and 14 (1 + 3i); here the cells are pretty similar to
honeycombs.
In the sequel we shall consider lattices of the form Λ = δZ + τ Z 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 pq built from our lattice, p, q ∈ Λ, and
p
=
q
(3.1)
with P =
ω1
δ p, Q
=
ω1
δ q
ω1
δ p
ω1
δ q
∈ Ω, where
Ω := ω1 Z + ω2 Z =
=
P
Q
ω1
ω1
(δZ + τ Z) =
Λ
δ
δ
ω2
by setting τ = δ ω
(which is not real by the linear independence of ω1 and ω2
1
over R and, hence, can be chosen as an element from the upper half-plane).
Complex Continued Fractions with Constraints on Their Partial Quotients
11
Therefore, any approximation by a quotient from Λ corresponds to an approximation by a quotient of the equivalent lattice Ω and vice versa. Lattices Ω1
and Ω2 are said to be equivalent if there exists a complex number ω 6= 0 such
that Ω1 = ωΩ2 .
Similarly to Hurwitz and (2.2) and (2.3), respectively, we consider a sequence of equations,
z = a0 +
1
,
z1
z 1 = a1 +
1
,
z2
... ,
z n = an +
1
zn+1
with an ∈ Λ chosen such that zn is in the Voronoı̈ cell VΛ (an ) around an ; of
course, the appearing zj are assumed not to vanish. This yields to a continued
fraction expansion
(3.2)
z = a0 +
1
1
1
1
= [a0 , a1 , . . . , an , zn+1 ]
...
a1 + a2 +
+ an + zn+1
with partial quotients in the lattice Λ. Obviously, a vanishing zn 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) := { qp : p, q ∈ Λ}, and the
continued fraction expansion for z is infinite.
In order to prove the convergence of this continued fraction expansion we
qn
and show the analogue of (2.4), i.e., |kn | > 1;
define once again kn = qn−1
here qn 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 Voronoı̈ cell
VΛ (0) ⊂ Dρ (0) with
(3.3)
ρ :=
1
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 an − zn ∈ VΛ (0) it follows that
zn+1 ∈ Dρ (0)−1 = C \ Dρ−1 (0).
Hence, zn+1 lies outside the disk of radius ρ−1 with center at the origin. In
order to prevent that zn+1 is located inside the Voronoı̈ cell VΛ (0) (which
would cause difficulties for convergence) we need to put a restriction on ρ. In
12
Hans Höngesberg, Nicola Oswald and Jörn Steuding
view of zn+1 ∈ VΛ (an+1 ) and |an+1 − zn+1 | ≤ ρ we obtain |an+1 | ≥ ρ−1 − ρ.
To conclude with the proof by induction we assume |kn | > 1 and deduce via
kn+1 = an+1 + k1n the inequality
1
> ρ−1 − ρ − 1
|kn |
√
which is greater than or equal to one for ρ ≤ 2 − 1. Hence,
|kn+1 | = |an+1 | −
Theorem. The continued fraction expansion (3.2)
√ with partial quotients in
the lattice Λ = δZ + τ Z converges provided ρ ≤ 2 − 1, 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
lattice Z[ 21 (1+ −3)] considered by Hurwitz [10]. A more sophisticated analysis
should lead to an extension of the above theorem covering these cases. Another,
more simple solution, relies on the observation (3.1) 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
in the middle (|τ | ≤
√ 3. The restrictions for ρ are indicated by the circle
√
2( 2 − 1)) and the neighbouring circles (|τ ± δ| ≤ 2( 2 − 1)) with the special
value δ = 12 . The set of admissible τ is given by the non-empty intersection of
all three circles.
In a similar way one could also consider arbitrary discrete point sets S in the
complex plane in place of a lattice provided the corresponding Voronoı̈ diagram
would share the essential property of having sufficiently small Voronoı̈ cells.
This would lead to another continued fraction expansion with partial quotients
Complex Continued Fractions with Constraints on Their Partial Quotients
13
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 2013 in honour
of Georgii Voronoı̈ and her kind hospitality.
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√
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14
Hans Höngesberg, Nicola Oswald and Jörn Steuding
[18] O. Perron, Die Lehre von den Kettenbrüchen, Teubner, Leipzig, 1st ed. 1913;
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Hans Höngesberg
Gunoldstraße 4/12, 1190 Vienna, Austria
[email protected]
Nicola Oswald
Department of Mathematics and Informatics, University of Wuppertal
Gaußstr. 20, 42 119 Wuppertal, Germany
[email protected]
and
Department of Mathematics, Würzburg University
Emil Fischer-Str. 40, 97 074 Würzburg, Germany
[email protected]
Jörn Steuding
Department of Mathematics, Würzburg University
Emil Fischer-Str. 40, 97 074 Würzburg, Germany
[email protected]