Journal of Number Theory 263 (2024) 153–205
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Journal of Number Theory
journal homepage: www.elsevier.com/locate/jnt
General Section
The worst approximable rational numbers
Boris Springborn
Technische Universität Berlin, Institut für Mathematik, MA 8-3,
Str. des 17. Juni 136, 10623 Berlin, Germany
a r t i c l e i n f o a b s t r a c t
Article history:
Received 24 July 2023
Received in revised form 19 March
2024
Accepted 1 April 2024
Available online 20 May 2024
Communicated by F. Pellarin
MSC:
10F20
30F60
Keywords:
Approximation constant
Diophantine approximation
Markov equation
Modular torus
We classify and enumerate all rational numbers with approx-
imation constant at least
1
3using hyperbolic geometry. Ra-
tional numbers correspond to geodesics in the modular torus
with both ends in the cusp, and the approximation constant
measures how far they stay out of the cusp neighborhood in
between. Compared to the original approach, the geometric
point of view eliminates the need to discuss the intricate sym-
bolic dynamics of continued fraction representations, and it
clarifies the distinction between the two types of worst ap-
proximable rationals: (1) There is a plane forest of Markov
fractions whose denominators are Markov numbers. They cor-
respond to simple geodesics in the modular torus with both
ends in the cusp. (2) For each Markov fraction, there are
two infinite sequences of companions, which correspond to
non-simple geodesics with both ends in the cusp that do not
intersect a pair of disjoint simple geodesics, one with both
ends in the cusp and one closed.
© 2024 The Author(s). Published by Elsevier Inc. This is an
open access article under the CC BY license (http://
creativecommons .org /licenses /by /4 .0/).
E-mail address: boris.springborn@tu-berlin.de.
https://doi.org/10.1016/j.jnt.2024.04.013
0022-314X/© 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC
BY license (http://creativecommons .org /licenses /by /4 .0/).
154 B. Springborn / Journal of Number Theory 263 (2024) 153–205
1. Introduction
For any real number x, the approximation constant
C(x)= inf
a
b∈Q\{x}b2·x−a
b,(1)
measures how well xcan be approximated by rational numbers a
b=x. Excluding xfrom
the admissible approximants a
bmakes C(x) strictly positive for all rational numbers x.
(This is easy to see and will be discussed shortly.) Intuitively, the larger the approxima-
tion constant C(x)of a rational number x ∈Q, the more isolated the rational number
xis among the other rational numbers.
An irrational number xis sometimes called badly approximable if C(x) >0, and this is
the case if and only if the sequence of partial denominators akof the continued fraction
expansion
x=a0+1
a1+1
a2+...
is bounded (see, e.g., [1, Prop. 1.32] or [19, §10.8]).
By contrast (as stated before), C(x) >0for all rational numbers x =p
q. Moreover,
the nonzero infimum in (1)is attained for some fraction a
bwith denominator b ≤q.
Because, on the one hand, if b >qand
a
b=p
q, then
b2·p
q−a
b=b
q·pb −qa>1.
On the other hand, if b =1and ais a closest integer to x(except xitself if x ∈Z) then
b2·x−a
b=|x−a|≤1
with equality if and only if x ∈Z.
This shows not only that all rational numbers are badly approximable in the sense that
C(x) >0, but also that the very worst approximable rational numbers are the integers
x ∈Zwith C(x) =1. They are followed by the half-integers x ∈Z +1
2with C(x) =1
2,
which are best approximated by the two nearest integers. This can be seen as a striking
confirmation of Hardy and Wright’s general observation: “From the point of view of ratio-
nal approximation, the simplest numbers are the worst” [19, p. 209, emphasis in original].
This article is about a geometric method to classify and enumerate the rational
numbers xwith approximation constant C(x) ≥1
3. Previously, the real numbers
with C(x) >1
3were classified in terms of their continued fraction representations by
Flahive [14]for rational xand by Gurwood [17]for irrational x. Flahive’s classification
apparently extends to include rational numbers with C(x) =1
3.
B. Springborn / Journal of Number Theory 263 (2024) 153–205 155
All of this is closely related to the classification of Markov irrationals and Markov
forms, i.e., of irrational numbers xwith Lagrange number L(x) >1
3and of indefinite
quadratic forms f(x, y) =Ax2+2Bxy +Cy2with Markov constant M(f) >2
3, where
L(x) = lim inf
b→∞ b2·min
a∈Zx−a
b(2)
and
M(f)= inf
(a,b)∈Z2\{(0,0)}
f(a, b)
√−det f.(3)
The monographs [1,6,10]and their bibliographies are excellent resources for that theory.
(Often the Lagrange number is defined as the reciprocal value, 1/L(x).)
While the Lagrange number L(x)and the Markov constant M(f)are invariant under
the actions of GL2(Z), the approximation number C(x)is only invariant under the group
of Z-affine transformations
x−→ ± x+n(n∈Z).(4)
In fact, the irrational numbers with C(x) >1
3classified by Gurwood are special repre-
sentatives of the GL2(Z)-classes of Markov irrationals. In this article, we focus on the
worst approximable rational numbers.
The connection between continued fractions and Diophantine approximation on the
one hand, and hyperbolic geometry on the other hand is well established [4,11,20]. Both
Markov forms and Markov irrationals correspond to geodesics in the modular torus [1,7,
8,16,18,32]: Markov forms correspond to simple closed geodesics, and Markov irrationals
correspond to simple geodesics with one end in the cusp and the other end spiraling into
a simple closed geodesic. The Lagrange number L(x)and the Markov constant M(f)
correspond to distances between geodesics and the Ford circles interpreted as horocycles.
This makes it possible to classify the Markov irrationals and Markov forms using purely
geometric methods [33]. In this article, we take the same geometric approach to classify
the worst approximable rational numbers.
Fig. 1illustrates the geometric interpretation of the approximation constant. The
extended real line RP1=R ∪{∞}is the ideal boundary of the hyperbolic plane H2
in the half-plane model. For an irrational number x ∈R \Qand a bound c >0, the
approximation constant satisfies the inequality C(x) ≥cif and only if xis visible from
∞behind the Ford circles after they are scaled by 2c. For a rational number, x ∈Q, the
condition is that xis visible after the Ford circle at xitself is removed, or equivalently,
that the highest point of the Ford circle at xis visible.
The classical approach to Diophantine approximation based on continued fractions
can also be interpreted geometrically, because continued fractions encode the symbolic
156 B. Springborn / Journal of Number Theory 263 (2024) 153–205
Fig. 1. Ford circles scaled by
2
3. Those whose highest point is visible from ∞belong to rational numbers
with approximation constant ≥1
3. A few of them are indicated in the figure (compare Fig. 3and Table 1).
The bottom figure shows a detail of the top figure.
dynamics of geodesics in the Farey triangulation. But our geometric approach goes fur-
ther. In effect, we trivialize the symbolic dynamics by considering not only one but all
ideal triangulations of the modular torus.
2. Overview
2.1. Classification of the worst approximable rational numbers
The main result of this article is the classification of the worst approximable rational
numbers in terms of Markov fractions and their companions. These special numbers can
be described in an elementary way as follows.
The forest of Markov fractions The Markov fractions between zero and one are naturally
associated with the faces of a plane binary tree. Its root edge and generating rule are
B. Springborn / Journal of Number Theory 263 (2024) 153–205 157
1
1
0
1
p2
q2p
q
p1
q1
p=p1q1+p2q2
p2q1−p1q2
q=q2
1+q2
2
p2q1−p1q2
(5)
Fig. 2. Root edge and generating rule for the tree of Markov fractions in [0, 1], of which one half is shown in
Fig. 3.
shown in Fig. 2. For example, the first branching after the root edge produces the Markov
fraction 1
2. At each node, the numbers pand qobtained by equations (5)are coprime
integers, so the Markov fraction
p
qis reduced.
Fig. 3shows the first few levels of the subtree containing the Markov fractions in the
interval [0,
1
2]. There is a symmetric subtree containing the Markov fractions in [1
2, 1],
obtained by a reflection and replacing each Markov fraction xwith 1 −x. Likewise, the
Markov fractions x +nin any other integer interval [n, n +1]are arranged in a similar
tree, and all these trees form a plane binary forest. The generating rule is the same for
all trees, and the root edges are infinite parallel rays separating faces associated with
consecutive integers (see Fig. 4).
Defining Markov fractions by this plane forest is simple but not very illuminating,
and it would be a bad starting point from which to develop the theory. So in Section 3.2,
we define the Markov fractions in terms of rational Markov triples (Definition 3.2) and
then construct the forest of Markov fractions in Lemma 3.8.
At each node in the plane forest, the three Markov fractions form a rational Markov
triple p1
q1,
p
q,
p2
q2. The three denominators q1, q, q2are Markov numbers forming a
Markov triple. The rational Markov triples at the vertices are also centered, i.e., the
middle denominator qis largest (see Definition 3.4). All centered rational Markov triples
arise in this way, except for the integer triples n −1, n, n +1∈Z3which correspond to
pairs of neighboring root edges. The best approximating rational numbers for the middle
Markov fraction
p
qare the two Markov fractions
p1
q1and
p2
q2(see Theorem 3.6).
The sequences of companions For each Markov fraction
p
q, there are two infinite se-
quences of companions, (γ+
k(p
q))k≥2and (γ−
k(p
q))k≥2defined as follows.
Definition 2.1. The right companions and left companions of a Markov fraction p
qare the
rational numbers γ+
k(p
q)and γ−
k(p
q)for k≥2, defined by
γ±
kp
q=p
q±uk−1
qu
k
,(6)
where (uk)is the integer sequence defined by the second order linear recursion
u0=0,u
1=1,u
k+1 =3qu
k−uk−1.(7)
158 B. Springborn / Journal of Number Theory 263 (2024) 153–205
0.5
2378
5741 0.4142135516. . .
408
985 0.4142131979. . .
206855
499393 0.4142128544. . .
70
169 0.4142011834. . .
3087111
7453378 0.4141895124. . .
6089
14701 0.4141895109. . .
529673
1278818 0.4141895093. . .
12
29 0.4137931034. . .
1354498
3276509 0.4133966975. . .
15571
37666 0.4133966972. . .
20226717
48928105 0.4133966970. . .
179
433 0.4133949191. . .
3472225
8399329 0.4133931412. . .
2673
6466 0.4133931333. . .
1
2
39916
96557 0.4133931253. . .
2
50.4
0
1
16725
43261 0.3866068745. . .
1120
2897 0.3866068346. . .
651838
1686049 0.3866067949. . .
75
194 0.3865979381. . .
1701181
4400489 0.3865890813. . .
2923
7561 0.3865890755. . .
113922
294685 0.3865890696. . .
5
13 0.3846153846. . .
19760
51641 0.3826416994. . .
507
1325 0.3826415094. . .
51709
135137 0.3826413195. . .
13
34 0.3823529411. . .
3468
9077 0.3820645587. . .
34
89 0.3820224719. . .
89
233 0.3819742489. . .
0.0
Fig. 3. Markov fractions in the interval [0,1
2]. Numerical values are shown in the right column.
B. Springborn / Journal of Number Theory 263 (2024) 153–205 159
.
.
.
n+1
–––– n+1
2
n
–––– n−
1
2
n−1
.
.
.
Fig. 4. Roots of the forest of Markov fractions.
Note that γ±
1p
q=p
qis the Markov fraction itself, so the sequences of companions
start with k=2(see also Remark 3.10).
The first companions γ±
2(p
q)are best approximated by the associated Markov frac-
tion p
qand no other rational number. All other companions, γ±
k(p
q)with k>2, are
best approximated by two rational numbers: the associated Markov fraction p
qand the
previous companion in the sequence, γ±
k−1(p
q)(see Theorem 3.14).
We can now state the main theorem:
Theorem 2.2. For a rational number
p
q∈Q, the following two statements are equivalent:
(i) C(p
q) ≥1
3
(ii) The number
p
qis either a Markov fraction or a companion of a Markov fraction.
Furthermore, C(p
q) =1
3if and only if xis the first left or right companion γ±
2(p
q)of a
Markov fraction x =p
q.
The proof of Theorem 2.2 is presented in Section 4.9. It relies on the characterization
of Markov fractions and their companions in terms of geodesics in the modular torus
(see Section 4).
Because a number-theoretic question deserves a number-theoretic answer, we first
define the Markov fractions and their companions algebraically in Section 3and then
derive the geometric characterization in Section 4. A more geometrically minded reader
might want to take the geometric characterization as the definition and read Sections 3
and 4in reverse order.
Yet another characterization is presented in Section 5, where Markov numbers and
their companions are described by a Fibonacci-like recursion along paths in a triangle
lattice. This represents the combinatorial point of view on Markov theory and illuminates
the link between Markov numbers and Christoffel words [28].
160 B. Springborn / Journal of Number Theory 263 (2024) 153–205
Interest in classifying the worst approximable rational numbers was initially motivated
by the uniqueness conjecture for Markov numbers [1,24,34]. This is the proposition —
yet to be proved or refuted — that every Markov number is the maximum of exactly
one ordered Markov triple. Indeed, the uniqueness conjecture for Markov numbers is
equivalent to the following uniqueness conjecture for Markov fractions:
Conjecture 2.3. For each Markov number q, the interval [0,
1
2]contains at most one
Markov fraction with denominator q.
One may equivalently write “exactly one” instead of “at most one”, because the
Markov fractions come in Z-affine orbits and every Markov number appears among
the denominators (see Section 3.2).
2.2. Limits of infinite paths in the forest
The forest structure of the set of Markov fractions is useful to study the set of its accu-
mulation points. But first note that every accumulation point of the set of Markov frac-
tions is irrational. Indeed, for
a
b∈Qand any Markov fraction p
q=a
b, we have p
q−a
b>
1
3b2because C(p
q) >1
3. So
a
bis not an accumulation point of the set of Markov fractions.
Now consider infinite paths in the tree of Markov fractions between 0and1, starting
at the root edge. Each path corresponds to a sequence of Markov fractions with increas-
ing denominators, which converges to an irrational number xwith Lagrange number
L(x) ≥1
3.
For example, always taking the lower branch in Fig. 3generates the sequence
1
2,2
5,5
13 ,13
34 , ...
which converges to
φ−2=1
23−√5=2−φ,
with φ =1
2(1 +√5), the golden ratio. This is a Markov irrational with Lagrange number
L(2 −φ) =L(φ) =1
√5>1
3.
Always taking the upper branch in Fig. 3generates the sequence
2
5,12
29 ,70
169 ,408
985 , ...
with limit −1 +√2. This is another Markov irrational, with Lagrange number L(−1 +
√2) =L(√2) =1
√8>1
3.
For these two examples, the limits represent the first two worst approximable classes
of irrational numbers. But there are only a countable number of PGL2(Z)-classes of
Markov irrationals and an uncountable number of paths in an infinite binary tree, so
B. Springborn / Journal of Number Theory 263 (2024) 153–205 161
1
1
1
2
2
2
5
5
5
5
5
5
29
29
29
29
29
29
13
13
13
13
13
13
169
169
169 169
169
169
433
433
433
433
433
433
194
194
194
194
194
194
34
34
34
34
34
34
q1
q2
q3
q1
q1
=3q2q3−q1=q22+q32
q1
Fig. 5. The tree of Markov numbers and its generating rule.
these two examples are exceptional. In fact, the limit xis a Markov irrational if the path
turns always left or always right except for finitely many branchings. If, on the other
hand, there are infinitely many left turns and infinitely many right turns, then the limit
has Lagrange number L(x) =1
3. Before we come back to this in Section 2.4, let us use
the Stern–Brocot tree to label Markov fractions.
2.3. Labeling the Markov fractions in the unit interval
The denominators of Markov fractions are Markov numbers. If we take the subtree of
Markov fractions in the interval [0,
1
2]shown in Fig. 3and replace the fractions by their
denominators alone, we obtain one of the six symmetric subtrees of the tree of Markov
numbers shown in Fig. 5. The combinatorial isomorphism with the Stern–Brocot tree
leads to the customary labeling of Markov numbers by the rational numbers in the
interval [0, 1]. If we do not forget about the numerators of the Markov fractions, we
obtain a labeling of the Markov fractions in the interval [0,
1
2]. For our purposes, it makes
sense to extend the Stern–Brocot tree to contain all non-negative rational numbers and
1
0=∞. That is, we consider the rooted trivalent plane tree with the following root edge
and generating rule:
1
0
0
1
p2
q2p1+p2
q1+q2
p1
q1
162 B. Springborn / Journal of Number Theory 263 (2024) 153–205
The combinatorial isomorphism of rooted plane trees leads to a bijection
μ:Q≥0∪{∞} −→ Markov fractions ∩[0,1]
n
m−→ μn
m.(8)
The symmetries of the (extended) Stern-Brocot tree and the tree of Markov fractions in
[0, 1] lead to the functional equation μn
m+μm
n=1. For example:
n
m
0
1
1
3
1
2
2
3
1
1
3
2
2
1
3
1
1
0
μn
m
0
1
5
13
2
5
12
29
1
2
17
29
3
5
8
13
1
1
The same correspondence μalso arises from the construction involving triangle paths
in the Eisenstein lattice explained in Section 5.
2.4. Limits of infinite paths (continued)
The combinatorial isomorphism between the (extended) Stern–Brocot tree and the
tree of Markov fractions in the interval [0, 1] extends to a bijection between infinite
paths starting at the root edges of either tree, respectively.
Each infinite path in the Stern–Brocot tree generates a sequence of nonnegative ra-
tional numbers that converges to a finite limit or diverges properly to ∞. The limit of a
path with infinitely many left turns and infinitely many right turns is irrational, and each
positive irrational number is the limit for precisely one such path. Indeed, the sequence
of partial denominators of the continued fraction expansion is the run-length encoding
of the sequence of left and right turns. On the other hand, an infinite path with only
finitely many left or finitely many right turns has a rational limit, and every positive
rational number is the limit for two such paths. The two corresponding paths in the tree
of Markov fractions have different but PGL2(Z)-equivalent Markov irrationals as limits.
Thus, the correspondence of infinite paths defines a map
μ:R≥0∪{∞} −→ PGL2(Z)-classes of real numbers with L≥1
3.
If xis rational, μ(x)is a class of Markov irrationals. Otherwise, μ(x)is a class of
irrationals with Lagrange number =1
3. It is not difficult to see that the restriction of μ
to Q ∩[0, 1] is a bijection onto the set of equivalence classes of Markov irrationals. What
about the restriction of μto [0, 1] \Q? The following two questions seem to be open:
Question 2.4. Is every irrational number xwith Lagrange number L(x) =1
3equivalent
to the limit of some infinite path in the tree of Markov fractions between 0and 1?
B. Springborn / Journal of Number Theory 263 (2024) 153–205 163
Table 1
The first few right companions, γ+
2, γ+
3, ..., of the Markov fractions
p
qwith denominator q<
1000. The limits are Markov irrationalities, i.e., irrational numbers xwith Lagrange number
L(x) >1
3. The fractions in the first row are equal to F2k−2/F2k, and the denominators in the
second row are equal to the Pell numbers P2k(see Remark 3.13 and compare Fig. 3).
p
q=γ+
1p
q,γ
+
2p
q,γ
+
3p
q,... −→ lim γkp
q
0
1,1
3,3
8,8
21 ,21
55 ,55
144 ,144
377 ,377
987 ,987
2584 , ... −→ 1
23−√5
1
2,7
12 ,41
70 ,239
408 ,1393
2378 ,8119
13860 ,47321
80782 , ... −→ 2−√2
2
5,31
75 ,463
1120 ,6914
16725 ,103247
249755 ,1541791
3729600 ,... −→ 1
10 19 −√221
5
13 ,196
507 ,7639
19760 ,297725
770133 ,11603636
30015427 , ... −→ 1
26 49 −√1517
12
29 ,1045
2523 ,90903
219472 ,7907516
19091541 ,687862989
1660744595 ,... −→ 1
58 111 −√7565
13
34 ,1327
3468 ,135341
353702 ,13803455
36074136 ,1407817069
3679208170 ,... −→ 1
17 32 −√650
34
89 ,9079
23763 ,2424059
6344632 ,647214674
1693992981 , ... −→ 1
178 335 −√71285
70
169 ,35491
85683 ,17993867
43441112 ,9122855078
22024558101 ,... −→ 1
338 647 −√257045
75
194 ,43651
112908 ,25404807
65712262 ,14785554023
38244423576 , ... −→ 1
97 183 −√21170
89
233 ,62212
162867 ,43486099
113843800 ,30396720989
79576653333 , ... −→ 1
466 877 −√488597
179
433 ,232522
562467 ,302045899
730644200 ,392357390279
949106253333 , ... −→ 1
866 1657 −√1687397
233
610 ,426391
1116300 ,780295297
2042828390 ,1427939967119
3738374837400 , ... −→ 1
305 574 −√209306
408
985 ,1205641
2910675 ,3562668747
8601043640 ,10527684941744
25416081045525 , ... −→ 1
1970 3771 −√8732021
Question 2.5. Do two different infinite paths exist in the subtree containing all Markov
fractions between 0and
1
2(see Fig. 3) such that their limits are PGL2(Z)-equivalent
with Lagrange number L =1
3?
2.5. Companions of Markov fractions
Table 1lists the first few right companions, γ+
2(p
q), γ+
3(p
q), ..., for each Markov frac-
tion
p
q=γ+
1(p
q)with denominator q<1000 (see Definition 2.1). The corresponding left
companions γ−
k(p
q)are symmetrically located:
γ−
kp
q+γ+
kp
q=2
p
q.(9)
The sequences of right and left companions converge to
p
q±δq, respectively, with
164 B. Springborn / Journal of Number Theory 263 (2024) 153–205
δq=3
2−9
4−1
q2.(10)
The closed interval
Ip
q=p
q−δq,p
q+δq(11)
contains all companions of
p
q, but neither any other Markov fractions nor their compan-
ions, but the endpoints of the interval are limit points of the set of Markov fractions (see
Lemma 4.16).
Fig. 6indicates a striking self-similarity in the arrangement of Markov fractions and
their companions.
2.6. Hyperbolic geometry
In the upper half-space model of the hyperbolic plane,
H2={z∈C|Im z>0}with metric ds =|dz|
Im z,
we associate a horocycle h(p, q)with every pair (p, q)of real numbers, not both zero:
•If q=0, h(p, q)is the horocycle centered at
p
qwith euclidean diameter
1
q2.
•h(p, 0) is the horizontal horocycle centered at ∞with equation Im z=p2.
This establishes a PSL2(R)-equivariant bijection between the space
(R2\{(0,0)})/{±1}
of lax vectors (in Conway’s terminology [9]) and the space of horocycles in the hyperbolic
plane [13, p. 665], [33, Ch. 5].
The following observation translates between Diophantine approximation and hyper-
bolic geometry:
Lemma 2.6 ([33, Proposition 8.1]). The signed distance d(gx, h(p, q)) of the vertical
geodesic gxin H2joining x ∈Rand ∞and a horocycle h(p, q)with q=0and
p
q=x
satisfies
d(gx,h(p, q)) = log 2q2x−p
q.(12)
Proof. See Fig. 7.
B. Springborn / Journal of Number Theory 263 (2024) 153–205 165
Fig. 6. Markov fractions and their companions. Markov fractions are marked by vertical lines and connected
to their companions by semicircles. The only companions that are visually discernible at the scales of these
figures are the first six right companions of
0
1(see Table 1). The vertical lines and semicircles project to
simple geodesics in the modular torus with both ends in the cusp.
166 B. Springborn / Journal of Number Theory 263 (2024) 153–205
xp
q
gx
h(p, q)
Im z=1
q2
Im z=2x−
p
q
d
d
Fig. 7. Signed distance d=d(gx,h(p, q)) between the vertical geodesic gxand the horocycle h(p, q).
If pand qare coprime integers, then the horocycle h(p, q)is a Ford circle. The previous
lemma relates the approximation number C(x)to the minimal signed distance between
the geodesic gxand a Ford circle h(p, q)not centered at either ∞or x(see also Fig. 1):
Corollary 2.7. For x ∈R,
log(2C(x)) = inf
p
q∈Q\{x}d(gx,h(p, q)).(13)
In particular, the following two statements are equivalent:
(i) C(x) ≥1
3
(ii) For all Ford circles h(p, q)with
p
q∈Q \{x},
d(gx,h(p, q)) ≥log 2
3.(14)
The hyperbolic plane H2is the universal cover of the modular torus M(see Sec-
tion 4.1), and the Ford circles project to the boundary of the maximal cusp neighborhood
in the modular torus. The geometric proof of the classification Theorem 2.2 for worst
approximable rational numbers in Section 4.9 also proves the following classification the-
orem of geodesics in Mthat have both ends in the cusp but otherwise stay furthest away
from the cusp:
Theorem 2.8. Let gbe a geodesic in the modular torus Mwith both ends in the cusp.
Then the following statements are equivalent:
(i) Away from the ends, the signed distance between gand the cusp is ≥log 2
3.
(ii) gdoes not intersect all simple closed geodesics in M.
(iii) gdoes not intersect all simple geodesics in Mwith both ends in the cusp.
If one and hence all of these conditions hold, then either
B. Springborn / Journal of Number Theory 263 (2024) 153–205 167
(a) gis a simple geodesic. In this case there is a unique simple closed geodesic that does
not intersect g, and infinitely many simple geodesics with both ends in the cusp that
do not intersect g.
(b) ghas self-intersections, and then there are a unique closed geodesic cin Mand a
unique simple geodesic ˜gin Mwith both ends in the cusp that do not intersect g.
In case (a), the geodesic gin Mis the projection of a vertical geodesic gxin H2for
some Markov fraction x. In case (b), the geodesic gin Mis the projection of a vertical
geodesic gxin H2for some companion xof a Markov fraction ˜x, and the geodesic ˜gin
Mis the projection of the vertical geodesic g˜xin H2.
3. The algebraic point of view
This section treats the two types of worst approximable rational numbers, the Markov
fractions and their companions, from the algebraic point of view. More geometrically
minded readers might want to read Section 4first, which provides a complementary
geometric point of view.
3.1. Markov numbers and Markov triples
For reference and to fix notation, we begin with a brief review of some basic definitions
and facts about Markov numbers and Markov triples [1,6,10].
A Markov triple (q1, q2, q3)is a positive integer solution of Markov’s equation
x2+y2+z2=3xyz . (15)
Equivalently, a triple of positive integers (q1, q2, q3) ∈(Z>0)3is a Markov triple if and
only if
q1
q2q3
+q2
q3q1
+q3
q1q2
=3.(16)
A Markov number is a positive integer that is an element of some Markov triple. For
example, (5, 1, 2) and (1, 5, 2) are two of the six Markov triples containing the Markov
numbers 1, 2, and 5.
Since Markov’s equation (15)is quadratic in each variable, there are two solutions for
each variable if the other two variables are fixed. Thus, if (q1, q2, q3)is a Markov triple,
then so are its neighbors
(q
1,q
2,q
3),(q1,q
2,q
3),(q1,q
2,q
3),
where the q
iare determined by Vieta’s formulas for quadratic polynomials, i.e.,
168 B. Springborn / Journal of Number Theory 263 (2024) 153–205
q
i=q2
j+q2
k
qi
=3qjqk−qi(17)
with {i, j, k} ={1, 2, 3}. This neighbor relation turns the set of Markov triples into a
plane binary tree. The Markov numbers correspond to the regions into which the tree
separates the plane (see Fig. 5).
The root Markov triple (1, 1, 1) and its neighbors (2, 1, 1), (1, 2, 1), and (1, 1, 2) are
called singular. All other Markov triples are called non-singular and consist of three
different Markov numbers. For each Markov triple, there is a unique path to the root. At
each step along this path, the largest Markov number of the triple is replaced according
to the rule (17).
3.2. Markov fractions and rational Markov triples
Markov fractions are one of the two types of worst approximable rational numbers
(see Theorems 2.2 and 3.6). Like the Markov numbers, we define Markov fractions in
terms of triples:
Definition 3.1. (i) A triple of rational numbers (x1, x2, x3)is a rational Markov triple if
xk=pk
qkfor some Markov triple (q1, q2, q3)and some integers p1, p2, p3satisfying the
equations
p2q1−p1q2=q3,(18)
p3q2−p2q3=q1.(19)
(ii) A Markov fraction is a rational number xkthat is an element of some rational
Markov triple (x1, x2, x3).
A rational Markov triple (x1, x2, x3)is strictly increasing,
x1<x
2<x
3,
because Markov numbers are positive and equations (18)and(19)are equivalent to
p2
q2−p1
q1
=q3
q1q2
and p3
q3−p2
q2
=q1
q2q3
.(20)
Equations (16), (17), and (20)also yield the relation
p3q1−p1q3=q
2(21)
with q
2defined by (17), which will be used in the proof of Lemma 3.2.
Since the Markov numbers qkof a Markov triple (q1, q2, q3)are pairwise coprime,
equations (18)and(19)imply that each pair pk, qkis also coprime, so the fractions
pk
qk
B. Springborn / Journal of Number Theory 263 (2024) 153–205 169
are reduced. This means that the Markov triple (q1, q2, q3)in Definition 3.1 is uniquely
determined by the rational Markov triple (x1, x2, x3). Further, if (x1, x2, x3)is a rational
Markov triple, then for every n ∈Z,
(x1+n, x2+n, x3+n)
is also a rational Markov triple with the same Markov triple (q1, q2, q3)of denominators,
and all rational Markov triples with this denominator triple are obtained in this way.
Since (−x3, −x2, −x1)is also a rational Markov triple, the group of Z-affine transfor-
mations (4)acts on the set of rational Markov triples, and hence on the set of Markov
fractions.
There is another Z-action on the set of rational Markov triples: Using equations (16)
and (20), it is easy to check that
(x3−3,x
1,x
2)and(x2,x
3,x
1+ 3 ) (22)
are also rational Markov triples, with cyclically permuted denominators. So the action of
the cyclic permutation group A3≃Z3on the set of Markov triples (q1, q2, q3) lifts to a
free action of Zon the set of rational Markov triples
p1
q1,
p2
q2,
p3
q3. Accordingly, the tree of
Markov numbers (see Fig. 5) corresponds to the forest of Markov fractions constructed
in Lemma 3.8, a subtree of which is shown in Fig. 3.
To see that there exists a rational Markov triple p1
q1,
p2
q2,
p3
q3for each Markov triple
(q1, q2, q3), note that the coprimality of Markov triples implies the existence of p1and
p2satisfying (18)(and p2and p3satisfying (19), and p1and p3satisfying (21)). By the
following lemma, there exists a suitable p3(or p1or p2, respectively) to complete the
rational Markov triple.
Lemma 3.2 (Completion of rational Markov triples). Let (q1, q2, q3)be a Markov triple.
(i) If p1, p2∈Zsatisfy equation (18), then
p2
2+1
q2∈Z, and
p3:=q1·p2
2+1
q2−p1p2(23)
satisfies equation (19).
(ii) If p2, p3∈Zsatisfy equation (19), then
p2
2+1
q2∈Z, and
p1:=p2p3−q3·p2
2+1
q2
(24)
satisfies equation (18).
170 B. Springborn / Journal of Number Theory 263 (2024) 153–205
(iii) If p1, p3∈Zsatisfy equation (21), then
p2:=p1q1+p3q3
q
2
(25)
is an integer and satisfies equations (18)and (19).
In any case, therefore, p1
q1,
p2
q2,
p3
q3is a rational Markov triple, which is, moreover,
uniquely determined by p1
q1,
p2
q2, p2
q2,
p3
q3, or p1
q1,
p3
q3, respectively.
For a proof see Section 3.2.1. The integrality
p2
2+1
q2implies the following necessary
condition for Markov numbers, which is well known [1, Lemma 3.14] and was proved in
a similar way [31, p. 196]:
Corollary 3.3. If qis a Markov number, then −1is a quadratic residue modulo q.
It will be useful to extend the distinction between singular and non-singular Markov
triples (see Section 3.1) to rational Markov triples, and to have a special term to single
out the rational Markov triples with denominators (1, 1, 1). Also, since we are especially
interested in rational Markov triples in which the second denominator is largest (see
Theorem 3.6), it makes sense to introduce a special term for those, too:
Definition 3.4. A rational Markov triple p1
q1,
p2
q2,
p3
q3is called
•singular if the Markov triple (q1, q2, q3)is singular, i.e., {q1, q2, q3} ⊂{1, 2},
•integral if q1=q2=q3=1, and
•centered if
q2≥max {q1,q
3}.(26)
Integral rational Markov triples are of the form (n −1, n, n +1)with n ∈Z. They
are integral, centered, and the only centered rational Markov triples for which (26)is
satisfied with equality. All non-integral singular Markov triples have one of the forms
(n, n +1
2, n +1), (n −2, n, n +1
2)or (n −1
2, n, n +2)for n ∈Z, and precisely those of
the first form are centered.
Remark 3.5. The notion of a centered rational Markov triple is closely related to Rock-
ett & Szüsz’s concept of a complete Markov triple [29, p. 103]: If
p1
q1,
p2
q2,
p3
q3is a centered
rational Markov triple with
pk
qk∈[0,
1
2]then (q2, p2; q1, p1; q3, p3)is a complete Markov
triple. Compare also Fig. 3with [29, p. 105]. However, the idea to interpret the pairs
(qi, pi)as rational numbers
pi
qidoes not seem to be obvious in the context of [29].
We can now state the main result about the approximation constant of a Markov
fraction, and the best approximating rationals that achieve it, as follows:
B. Springborn / Journal of Number Theory 263 (2024) 153–205 171
Theorem 3.6 (Best approximants of Markov fractions). (i) Every Markov fraction is the
middle element of a unique centered rational Markov triple.
(ii) If p1
q1,
p2
q2,
p3
q3is a centered rational Markov triple, then the best approximants
of p2
q2are precisely p1
q1and p3
q3, i.e.,
Cp2
q2=q2
1·p2
q2−p1
q1=q2
3·p3
q3−p2
q2,
and for every rational number
a
bnot contained in the triple,
Cp2
q2<b
2·
p2
q2−a
b.
In particular, this implies
Cp2
q2=q1q3
q2
>1
3.(27)
For a proof of (i) see Section 3.2.3, for a proof of (ii) see Section 4.6.
Examples 3.7. Since 0
1,
2
5,
1
2and 2
5,
12
29 ,
1
2are centered rational Markov triples,
(i) the best approximants of
2
5are
0
1and
1
2, and C2
5=2
5,
(ii) the best approximants of
12
29 are
2
5and
1
2, and C12
29 =10
29 .
Fig. 10 illustrates the geometric interpretation of these examples.
The plane forest of centered rational Markov triples (see Section 2.1 and Fig. 3) is con-
structed in the following lemma. It is used in Section 3.2.3 for the proof Theorem 3.6 (i)
and in Section 4.2 to establish the geometric characterization of rational Markov triples.
Lemma 3.8 (Forest of centered rational Markov triples).
(i) If p1
q1,
p2
q2,
p3
q3is a rational Markov triple, then so are its children
p1
q1
,p
3
q
3
,p2
q2and p2
q2
,p
1
q
1
,p3
q3(28)
and its parents
p
2
q
2
,p1
q1
,p3
q3,p1
q1
,p3
q3
,p
2
q
2
+3
(29)
where q
1, q
2, and q
3are defined by (17), and p
1, p
2, and p
3defined by
172 B. Springborn / Journal of Number Theory 263 (2024) 153–205
p
1=1
q1
(p2q2+p3q3),(30)
p
2=p3p1−p2
1+1
q1
q3,(31)
p
3=1
q3
(p1q1+p2q2).(32)
In particular, p
k∈Zfor k∈{1, 2, 3}.
(ii) A rational Markov triple is a child of each of its parents and a parent of each of its
children.
(iii) Both children of a centered rational Markov triple are centered, and exactly one
parent of a non-singular centered rational Markov triple is centered.
Thus, the parent-child relationship induces on the set of non-integer centered rational
Markov triples the structure of a plane binary forest, with singular root triples (n, n +
1
2, n +1)as shown in Fig. 4and the generating rule (5).
For a proof, see Section 3.2.2.
3.2.1. Proof of Lemma 3.2 (completion of rational Markov triples)
To see part (i) of Lemma 3.2, first note that q2divides p2
2+ 1 because q2and q1are
coprime and
q2
1(p2
2+1) (18)
=(p1q2+q3)2+q2
1
(15)
=q2(p2
1q2+2p1q3+3q1q3−q2).
A straightforward calculation yields equation (19):
p3q2−p2q3
(23)
=p2(p2q1−p1q2)+q1−p2q3
(18)
=q1
Statement (ii) can be proved in the same way.
For the integrality statement of (iii), consider the identity
q1(p1q1+p3q3)(21)
=p1(q2
1+q2
3)+q
2q3
(17)
=q
2(p1q2+q3).
So q
2divides (p1q1+p3q3) because q1and q
2are coprime as elements of the Markov
triple (q1, q
2, q3).
To verify equations (18)and(19), use the equivalent forms (20): With
p2
q2
(25)
=p1q1+p3q3
q2q
2
(17)
=p1q1+p3q3
q2
1+q2
3
we obtain
p2
q2−p1
q1
(25)
=(p3q1−p1q3)q3
(q2
1+q2
3)q1
(21)
=q
2q3
(q2
1+q2
3)q1
(17)
=q3
q1q2
,
B. Springborn / Journal of Number Theory 263 (2024) 153–205 173
proving (18). An analogous calculation proves (19).
Finally, since it has been established that p1
q1,
p2
q2,
p3
q3is a rational Markov triple, the
uniqueness of the completion follows from equations (20).
3.2.2. Proof of Lemma 3.8 (forest of centered rational Markov triples)
Parts (i) and (ii) of Lemma 3.8 can be proved by applying the Completion-Lemma 3.2
appropriately. For example, to see that the first child is a rational Markov triple, let
˜p1
˜q1=p1
q1and
˜p3
˜q3=p2
q2. Then
˜p3˜q1−˜p1˜q3=q3=: ˜q
2,
and Lemma 3.2 (iii) says that
˜p1
˜q1
,˜p2
˜q2
,˜p3
˜q3=p1
q1
,p
3
q
3
,p2
q2
is a rational Markov triple.
Statement (iii) follows from the same properties of Markov triples and their neighbors
that produce the tree of Markov numbers [6]: If q2≥max{q1, q3}, then q
1>q
2and
q
3>q
2, so both children of a centered rational Markov triple are centered. If q2>
max{q1, q3} >1, then min{q1, q3} ≤q
2<max{q1, q3}, so exactly one parent of a non-
singular centered rational Markov triple is centered.
Therefore, starting with any non-singular centered rational Markov triple and travers-
ing up the parental line along centered triples, one will eventually reach a singular root
triple (n, n +1
2, n +1).
The generating rule (5)is equivalent to the equations for children.
3.2.3. Proof of Theorem 3.6 (i). Every Markov fraction is the middle element of a
unique centered triple
By definition, every Markov fraction
p
qis contained in some rational Markov triple. If q
is not the largest denominator in that triple, iteratively replace the triple with the parent
or child in which the fraction with largest denominator is removed until qis the largest
denominator. If
p
qis not the middle element in this triple then it is the middle element in
one of the adjacent triples (22). This shows that there exists a centered rational Markov
triple mwith
p
qin the middle.
To see that mis the unique centered rational Markov triple with
p
qin the middle, con-
sider first the tree of centered rational Markov triples that contains m. All descendants of
mhave a larger denominator in the middle, and all ancestors have a smaller denominator
in the middle. So no other centered triple in the same tree has
p
qin the middle. But each
tree in the forest of centered rational Markov triples contains only Markov fractions in
a unique integer interval [n, n +1]. So if a Markov fraction is contained in two centered
rational Markov triples of different trees, then it is an integer, and it is the last element
174 B. Springborn / Journal of Number Theory 263 (2024) 153–205
in one triple and the first element in the other. This shows that no two centered rational
Markov triples have the same middle element.
3.3. Companions of Markov fractions
This section deals with the companions of Markov fractions (see Definition 2.1) from
the algebraic point of view. Proofs in which we use the geometric interpretation are
deferred to Section 4.
We will often use the following lemma implicitly (e.g., in the statement of Theo-
rem 3.14).
Lemma 3.9. The common denominator on the right hand side of equation (6)is indeed
qu
k, so the fraction representation
γ±
kp
q=pu
k±uk−1
qu
k
is reduced.
Proof. See Corollary 4.7.
Remark 3.10 (Index issues). For k=1, equation (6)says
γ±
1p
q=p
q,(33)
But Definition 2.1 explicitly requires k≥2, so the Markov fraction
p
qis by definition
not a companion of itself. Consequently, the k-th left and right companions are γ+
k+1(p
q)
and γ−
k+1(p
q), respectively. One could fix this notational nuisance either by changing the
definition and counting the Markov fractions as their own companions, or by shifting
the index in the definition of γ±
k. The first option is bad because Markov fractions
and their companions are different both regarding Diophantine approximation (compare
Theorems 3.6 and 3.14) and from the geometric point of view (compare Lemmas 4.2
and 4.4). The second option is bad because it would create other nuisances. For example,
the symmetries (34)and(35)of the following remark would become more complicated.
Remark 3.11 (Symmetry). The recursion (7)can be used to define ukand hence γ±
knot
only for k≥0 but for all k∈Z. Then
u−k=−uk(34)
and this implies γ±
0p
q=∞and the following symmetry relating γ+and γ−:
γ+
−kp
q=γ−
kp
q+3
(35)
B. Springborn / Journal of Number Theory 263 (2024) 153–205 175
The following properties of the sequence (uk)k∈Zare easily verified by induction.
Lemma 3.12 (Properties of uk). For any Markov number q, the sequence (uk)k∈Zdefined
by (7)is
(i) strictly increasing and
(ii) satisfies the equation
u2
k+1 −3qu
k+1 uk+u2
k= 1 (36)
for all k∈Z.
Remark 3.13 (Fibonacci and Pell numbers). (i) For q=1,
uk=F2kand so γ±
k(p)=p±F2k−2
F2k
,
where (Fk)is the sequence of Fibonacci numbers,
F0=0,F
1=1,F
k+1 =Fk+Fk−1.
(ii) For q=2and odd p,
2uk=P2kand so γ±
kp
2=p
2±P2k−2
2P2k
,
where (Pk)is the sequence of Pell numbers,
P0=0,P
1=1,P
k+1 =2Pk+Pk−1.
Theorem 3.14 (Best approximants of companions). The best approximants of a compan-
ion γ±
kp
q, k≥2, are precisely
p
qand γ±
k−1p
q, i.e.,
Cγ±
kp
q=q2γ±
kp
q−p
q=(qu
k−1)2γ±
kp
q−γ±
k−1p
q(37)
and
Cγ±
kp
q<b
2γ±
kp
q−a
b
for every rational number
a
bexcept
p
q, γ±
k−1p
q, and γ±
kp
q. Moreover,
Cγ±
kp
q=quk−1
uk≥1
3,(38)
where equality holds if and only if k=2, i.e., for first companions.
176 B. Springborn / Journal of Number Theory 263 (2024) 153–205
Fig. 8. Fundamental domain of the modular torus with gluing transformations.
For a proof, see Section 4.7. The first companions γ±
2(p
q)with approximation constant
C=1
3are therefore the best of the worst approximable rational numbers. They are also
distinguished by the property that the approximation constant is attained for one unique
best approximant,
p
q=γ±
1p
q, instead of two.
Examples 3.15. (i) The first right and left companions of the Markov fraction 1
2are
γ+
2(1
2) =7
12 (compare Table 1) and γ−
2(1
2) =5
12 . They are best approximated by their
associated Markov fraction
1
2and no other rational number. The approximation constant
is C(7
12 ) =C(5
12 ) =1
3.
(ii) The second right and left companions of
1
2are γ+
3(1
2) =41
70 (compare Table 1)
and γ−
3(1
2) =29
70 . They are best approximated by exactly two rational numbers: their
associated Markov fraction
1
2and its first right and left companion, respectively. The
approximation number is C(41
70 ) =C(29
70 ) =12
35 .
4. The geometric point of view
4.1. The modular torus
Let G <PSL2(Z)be the commutator subgroup of the modular group PSL2(Z). The
group Gis freely generated by the two generators
[A]=1−1
−12
and [B]=11
12
.
It is a normal subgroup of PSL2(Z)with index 6, and it is the only subgroup that has
a once punctured torus as orbit space. This orbit space M=G\H2with canonical
projection
π:H2−→ M, π(z)=Gz.
is called the modular torus. The ideal quadrilateral with vertices −1, 0, 1, and ∞is a
fundamental domain, from which one obtains the torus Mby gluing the pairs of opposite
sides via the isometries [A]and[B], respectively (see Fig. 8).
B. Springborn / Journal of Number Theory 263 (2024) 153–205 177
Fig. 9. The Farey triangulation. Any two adjacent triangles form a fundamental domain of G. Edges labeled
with the same letter project to the same edge in the modular torus M. An element of PSL2(Z) belongs to
Gif and only if it respects edge labels and triangle shadings. An element of PSL2(Z) corresponds to the
hyperelliptic involution of Mif and only if it respects edge labels but maps shaded triangles to unshaded
ones and vice versa.
The Farey triangulation of the projective plane H2projects to the most symmetric
ideal triangulation of M(see Fig. 9). The Ford circles (the unshaded circles in Fig. 1)
project to the boundary of the largest embedded cusp neighborhood in M.
4.2. The geometric characterization of rational Markov triples and Markov fractions
Rational Markov triples correspond to the fundamental domains of the modular
torus Mthat are ideal quadrilaterals with one vertex at ∞. This is illustrated in Fig. 8for
the triple (−1, 0, 1), in Fig. 10 for the triples (0
1,
2
5,
1
2)and (2
5,
12
29 ,
1
2), and schematically
in Fig. 11.
Remark 4.1. For the sake of clearer pictures, Figs. 11, 13, and 15 do not show the correct
geometry of the modular torus Mbut some other hyperbolic tori with one cusp. As a
result, the objects in these figures are more evenly sized and intersect each other less
than would be the case in the modular torus.
To put the correspondence between rational Markov triples and fundamental domains
more precisely, we formulate the following lemma:
Lemma 4.2 (Geometric characterization of rational Markov triples). The following state-
ments for real numbers x1<x
2<x
3are equivalent:
(i) (x1, x2, x3)is a rational Markov triple.
(ii) The ideal quadrilateral with vertices x1, x2, x3, ∞is a fundamental domain of the
modular torus M.
(iii) The ideal triangles with vertices x1, x2, ∞and x2, x3, ∞, respectively, project to the
two triangles of some ideal triangulation of the modular torus M.
(For a proof, see Section 4.3.)
Since every simple geodesic with both ends in the cusp is an edge of some ideal triangu-
lation of the modular torus M, we get the following characterization of Markov fractions:
178 B. Springborn / Journal of Number Theory 263 (2024) 153–205
Fig. 10. Fundamental domains corresponding to the centered rational Markov triples
0
1,
2
5,
1
2and
2
5,
12
29 ,
1
2
of Example 3.7. The approximation constants of
2
5and
12
29 are e1
2d=C2
5=2
5and e1
2d=C12
29 =10
29 ,
respectively (see Lemma 2.6 and Corollary 2.7). The signed distances dand dare negative because the
vertical geodesics at
2
5and
12
29 intersect the Ford circles at the best approximants.
Corollary 4.3 (Geometric characterization of Markov fractions). For a real number x ∈
R, the following statements are equivalent:
(i) xis a Markov fraction.
B. Springborn / Journal of Number Theory 263 (2024) 153–205 179
Fig. 11. If (p1
q1,
p2
q2,
p3
q3)is a rational Markov triple, then the ideal quadrilateral with vertices
p1
q1,
p2
q2,
p3
q3,
and ∞is a fundamental domain of the modular torus. Dashed: If we cut along the diagonal q2and glue
the left ideal triangle
p1
q1,
p2
q2, ∞back to the right ideal triangle
p2
q2,
p3
q3, ∞, but along edge q3, we obtain
another fundamental domain, which corresponds to the rational Markov triple (p2
q2,
p3
q3,
p1
q1+ 3) (compare
equation (22)).
(ii) The vertical geodesic gxjoining xand ∞in H2projects to a simple geodesic π(gx)
in the modular torus Mwith both ends in the cusp.
4.3. Proof of Lemma 4.2 (geometric characterization of rational Markov triples)
The equivalence of statements (ii) and (iii) of the lemma is obvious. It remains to
show that (i) and (iii) are equivalent.
(iii)⇒(i) Assuming (iii) implies that the vertices xk∈Rare rational,
xk=pk
qk∈Q,
because they project to the ideal point of the modular torus.
The denominators qkare equal to the weights of the edges of the triangulation [33,
Ch. 11] (see Fig. 11). Indeed, the euclidean diameter of the Ford circle at
pk
qkis q−2
k, so
the distance to the horocycle {Im z=1}is dk=2 log qk, and the weight is qk=edk/2.
By Penner’s geometric interpretation of Markov’s equation [25,26], the denominators
form a Markov triple (q1, q2, q3). Finally, the relation between the edge weights and
the horocyclic arcs of a decorated ideal triangle (see Fig. 12) implies equations (20).
Therefore, (x1, x2, x3) =p1
q1,
p2
q2,
p3
q3is a rational Markov triple.
(i)⇒(iii) The implication is true for any integer rational Markov triple, i.e., if
(x1, x2, x3) =(n −1, n, n +1)for some n ∈Z. We will show: If the implication is true
for a rational Markov triple p1
q1,
p2
q2,
p3
q3, then it also holds for the adjacent triples (22)
and the children (28). Since every centered rational Markov triple is a descendant of an
180 B. Springborn / Journal of Number Theory 263 (2024) 153–205
Fig. 12. Ideal triangle decorated with
horocycles. Relation between edge
weights qkand horocyclic arcs.
Fig. 13. Fundamental domain consist-
ing of ideal triangles
p1
q1,
p
3
q
3, ∞and
p
3
q
3,
p2
q2, ∞.
integer triple, and every non-centered rational Markov triple is adjacent to a centered
one, this proves the implication (i)⇒(iii) for all rational Markov triples.
So let
p1
q1,
p2
q2,
p3
q3be a rational Markov triple for which the implication (i)⇒(iii) holds.
To see that the same is true for the adjacent triples (22), note that z→ z+6is a deck
transformation of the modular torus, and z→ z+3 is a lift of the hyperelliptic involution.
It remains to show that the implication (i)⇒(iii) also holds for the children (28). We
will present the argument for the left child p1
q1,
p
3
q
3,
p2
q2, see Fig. 13. The argument for
the right child is analogous.
First we choose a matrix S∈SL2(R)that satisfies
Sp1
q1=±p2
q2and Sp2
q2=±p1
q1,
which determines Sup to sign. Let us choose
S=p2p1
q2q1·−p1p2
−q1q2−1
=1
p2q1−p1q2p1q1+p2q2−p2
1−p2
2
q2
1+q2
2−p1q1−p2q2.
Then the hyperbolic isometry [S] ∈PSL2(R)maps the Ford circle h(p1, q1)at
p1
q1to
the Ford circle h(p2, q2)at
p2
q2and vice versa (see Section 2.6). This implies that [S] ∈
PSL2(R)is a lift of the hyperelliptic involution of the modular torus M, and therefore
S∈SL2(Z).
Since the ideal triangle with vertices
p1
q1,
p2
q2, ∞is a lift of one triangle of an ideal
triangulation of the modular torus, the ideal triangle with vertices
[S]·p1
q1
=p2
q2
,[S]·p2
q2
=p1
q1
,and [S]·∞
B. Springborn / Journal of Number Theory 263 (2024) 153–205 181
Fig. 14. Topological sketch of the modular torus Mshowing the geodesic π(gx) corresponding to a Markov
fraction x, and the simple closed geodesic ηthat does not intersect it. The geodesic π(gγ+
2(x)) corresponds
to the first right companion γ+
2(x). It is contained in the cylinder π(Rx) bounded by π(gx)and ηand has
one self-intersection.
is a lift of the other triangle of the triangulation. (Here ·denotes the isometric action of
PSL2(R)on H2.) Applying an edge flip, we obtain a triangulation of the modular torus
whose triangles lift to the ideal triangles with vertices
p1
q1
,S·∞,∞and S·∞,p2
q2
,∞,
respectively. Finally, using equation (18)and (q2
1+q2
2)/q3=q
3, we obtain
S1
0=p
3
q
3,so S·∞=p
3
q
3
and the implication (i)⇒(iii) holds for the child p1
q1,
p
3
q
3,
p2
q2, too. This completes the
proof of Lemma 4.2.
4.4. The geometric characterization of companions
Figs. 14 and 15 illustrate the geometric interpretation of the first right compan-
ion γ+
2(x).
Generally, the vertical geodesic gxprojects to a simple geodesic π(gx)in the Modular
torus Mwith both ends in the cusp (by Corollary 4.3). Then there is a unique simple
closed geodesic ηin Mthat does not intersect π(gx). The geodesics π(gx)and ηseparate
Minto two cylinders, which are labeled π(Rx)and π(Lx)in Fig. 14 for reasons that will
become clear in the following Section 4.5. Be that as it may, π(Rx)is the cylinder to the
right of π(gx)if gxis oriented in the direction from xto ∞, and π(Lx)is the cylinder to
the left.
182 B. Springborn / Journal of Number Theory 263 (2024) 153–205
Fig. 15. The right companions γ+
2(x), ... are ideal vertices of region Rxbetween xand x +3
2. The left
companions γ−
2(x), ... are ideal vertices of the region Lxbetween x −3
2and x. The dotted lines are the
geodesics [T]k·gγ+
2(x).
Now let ybe a right or left companion of x, i.e., y=γ+
k(x)or y=γ−
k(x)for some
k≥2. Then the vertical geodesic gyprojects to a non-simple geodesic π(gy)in the
cylinder π(Rx)or π(Lx), respectively, with k−1 self-intersections.
Formulating the converse statement is slightly more involved. To understand why, note
first that the isometry z→ z+6 of H2is a deck transformation of M(see Fig. 9). Hence
the companions of xcorrespond to the same vertical geodesics in Mas the companions of
x +6nfor any n ∈Z. Similarly, the isometry z→ z+3 of H2projects to the hyperelliptic
involution of M, which reverses the orientations of π(gx)and η. As a consequence, the
cylinder π(Rx)contains not only the projected vertical geodesics for the right companions
γ+
k(x), but also those for γ−
k(x +3), the left companions of x +3 (see Fig. 15 and compare
Remark 3.11).
Altogether, the following lemma characterizes the right and left companions of a
Markov fraction geometrically:
Lemma 4.4 (Geometric characterization of companions). Let xbe a Markov fraction, so
that π(gx)is a simple geodesic in the modular torus Mwith both ends in the cusp. Let
ηbe the unique simple closed geodesic in Mthat does not intersect π(gx). Then for a
number y∈Rand the vertical geodesic gy, the statements (i+) and (ii+) are equivalent,
and the statements (i−) and (ii−) are equivalent:
(i+)yis a right companion of x, i.e., y=γ+
k(x)for some k≥2.
(ii+)y∈[x, x +3
2]and π(gy)is a geodesic with both ends in the cusp that intersects
neither π(gx)nor η.
(i−)yis a left companion of x, i.e., y=γ−
k(x)for some k≥2.
(ii−)y∈[x −3
2, x]and π(gy)is a geodesic with both ends in the cusp that intersects
neither π(gx)nor η.
B. Springborn / Journal of Number Theory 263 (2024) 153–205 183
The geodesics π(gγ+
k(x))and π(gγ−
k(x))have k−1self-intersections.
4.5. Proof of Lemma 4.4 (geometric characterization of companions)
The geodesics in the G-orbit G ·gxof the vertical geodesic gxtogether with π−1(η), the
G-orbit of lifts of the closed geodesic η, separate the hyperbolic plane H2into simply
connected domains. Each one of these domains covers one of the two cylinders into
which π(gx)andηseparate M. Of these domains, let Rxand Lxbe the ones to the right
and to the left of gx, respectively, when gxis oriented in the direction from xto ∞(see
Fig. 15). In short, Rxand Lxare the domains to the left and right of gxsuch that the
restrictions π|Rxand π|Lxare universal covers of the two cylinders in M.
The geometric characterization of the companions γ±
k(x) follows in a straightforward
manner from an explicit analytic description of the domains Rxand Lx. We will perform
the calculations only for Rxand the right companions γ+
k(x). The domain Lxand the
left companions γ−
k(x)can be treated analogously. Alternatively, one may use the fact
that z→ z+ 3 projects to the hyperelliptic involution to see that Lx=Rx−3(compare
Remark 3.11).
In order to describe the boundary of the domain Rx, note first that the cylinder
π(Rx) ⊂Mhas two geodesic boundary components. One is the closed geodesic η, and the
other is the geodesic π(gx)with both ends in the cusp of M. Accordingly, the boundary
of the universal cover Rxof this cylinder consists of two parts (compare Fig. 15):
(1) a hyperbolic line xthat is the universal cover of the closed geodesic η, and
(2) a doubly infinite sequence of hyperbolic lines in the G-orbit of gxincluding gxitself,
that form a doubly infinite ideal polygon.
Since π(Rx)is a cylinder with fundamental group Z, the group of deck transformations
of the universal cover Rx→π(Rx)is generated by a hyperbolic isometry
[T]∈G⊂PSL2(Z),
which maps the hyperbolic line xto itself and each edge of the ideal polygon (2) to the
next. In particular, [T]is a hyperbolic translation with axis x, the edges of the ideal
polygon are [T]k·gxfor k∈Z, and therefore the ideal vertices of the polygonal boundary
component accumulate at the endpoints of the axis x.
One geodesic in the polygonal boundary of Rxis known by definition, the vertical
geodesic gx. Another one is the vertical geodesic gx+3. To see this, note that since π(gx)
is simple geodesic in the modular torus Mwith both ends in the cusp, there is an ideal
triangulation of Mconsisting of π(gx)and two other such geodesics. Now contemplating
Fig. 11 with x =p1
qq, one sees that gxand gx+3 are adjacent edges in the ideal polygonal
boundary of Rxsharing the common vertex ∞.
184 B. Springborn / Journal of Number Theory 263 (2024) 153–205
To derive an explicit expression for the generator [T] ∈PSL2(Z), we let
x=p
q,
and we may assume that [T]maps gx+3 to gx. (The other generator is [T]−1, which maps
gxto gx+3.) This means not only that [T]maps the endpoints of gx+3 (x +3and ∞) to
the corresponding endpoints of gx(∞and x). But the isometry [T]also maps the Ford
circles at these corresponding endpoints to each other, i.e., h(p +3q, q)to h(1, 0) and
h(1, 0) to h(p, q), which determines the matrix T∈SL2(Z)up to sign (see Section 2.6
and [33, Ch. 5]). Choosing
T1p+3q
0q=−p1
−q0,
which works because
det 1p+3q
0q=q=det
−p1
−q0
we obtain
T=−p3p+p2+1
q
−q3q+p∈SL2(Z).(39)
Remark 4.5. In particular, this implies
p2+1
q∈Z, which provides a geometric interpre-
tation of the number theoretic Corollary 3.3. More directly, such a geometric interpre-
tation can be obtained by considering the orientation preserving hyperbolic isometry
[Sp,q] ∈PSL2(R)that interchanges the horocycles h(p, q)and h(1, 0). This determines
Sp,q =−pp2+1
q
−qp
∈SL2(R)
up to sign. Since [Sp,q] projects to the hyperelliptic involution on M, we have Sp,q ∈
SL2(Z). (The map [T]is the composition of [Sp,q] followed by z→ z+3.)
The axis xof the hyperbolic translation [T] connects the attracting and repelling
fixed points of [T]. Let us call them
ξ+= lim
k→+∞[T]k·∞and ξ−= lim
k→−∞[T]k·∞,(40)
respectively, where convergence is in the topology of CP1. (We might equally well let
[T]kact on any other point except the fixed points, but we choose ∞so we can re-use
equation (40)in the proof of Lemma 4.16.) By a straightforward calculation we obtain
B. Springborn / Journal of Number Theory 263 (2024) 153–205 185
ξ±=x+3
2∓9
4−1
q2.(41)
Remark 4.6. Since trace(T) =3q, the eigenvalues of Tare
1
2(3 q±9q2−4). This also
shows the well-known formulas for the lengths of closed geodesics in the modular torus,
2cosh
length(η)
2=3q, 2exp
length(η)
2=3q+9q2−4,
but we will not need them.
Now the equivalence of the statements (i+) and (ii+) follows from the fact that
[T]k·∞=γ+
k(x) (42)
(with γ+
k(x) defined by equation (6)). To see this, let (rk)k∈Zand (sk)k∈Zbe defined by
rk
sk=Tk−1
0,(43)
so that
[T]k·∞=rk
sk
.
Then it suffices to show that
rk
sk=pu
k+uk−1
qu
k(44)
for the sequence (uk)k∈Zdefined by (7).
First note that qdivides skfor all k∈Z. (This is easy to see by induction in both
directions starting from s0=0.) Hence we can define integer sequences (tk)k∈Zand
(uk)k∈Zby
uk=sk
qand tk=rk−pu
k.
We have to show that (uk)k∈Zsolves the recursion (7). The initial conditions are easily
verified. Using the relation
rk
sk=1p
0q
=: U
tk
uk,
and rk+1
sk+1 =Trk
skwe obtain the first order recursion formula
186 B. Springborn / Journal of Number Theory 263 (2024) 153–205
tk+1
uk+1 =01
−13q
U−1TU
tk
uk.
Eliminate (tk)using tk=uk−1to obtain the second-order recursion (7)for (uk)k∈Zand
equation (44)for rkand sk. This completes the proof of equation (42), showing that (i+)
and (ii+) are equivalent.
Finally, to see that the geodesic π(gγ+
k(x))in the cylinder π(Rx) ⊂Mhas k−1self-
intersections, note that the vertical geodesic gγ+
k(x)in the universal cover Rxintersects
the 2 (k−1) geodesics
[T]·gγ+
k(x)for ∈{1,...,k−1}∪{−(k−1),...,−1}.(45)
but the intersection points
ζ=gγ+
k(x)∩[T]·gγ+
k(x)(46)
project in pairs to the same point in M,
π(ζ)=π(ζ−),(47)
because
ζ=[T]·ζ−.(48)
This completes the proof of Lemma 4.4.
Because T∈SL2(Z), equation (43) implies that rkand skare coprime for all k∈Z.
So the above proof (together with the analogous argument for the left companions) yields
the following corollary:
Corollary 4.7. For all k∈Z, the integers p uk+uk−1and qu
kare coprime, and so are
the integers p uk−uk−1and qu
k.
4.6. Proof of Theorem 3.6 (ii) (best approximants of Markov fractions)
Part (i) of Theorem 3.6 was proved without recourse to geometry in Section 3.2.3.
The purpose of this section is to prove part (ii). The argument begins algebraically, but
geometry will come into play shortly.
Let
m=p1
q1
,p2
q2
,p3
q3
be a centered rational Markov triple. Then we have
B. Springborn / Journal of Number Theory 263 (2024) 153–205 187
q2
1·p2
q2−p1
q1(18)
=q1q3
q2
(19)
=q2
3·p3
q3−p2
q2
.,
Moreover,
1
3<q1q3
q2≤1
from equation (16), together with (26)for the upper bound. It remains to show that for
every rational number
a
b∈ p1
q1,
p2
q2,
p3
q3,
q1q3
q2
<b
2·
p2
q2−a
b.(49)
To see this, we will rely on the geometric characterization of Markov fractions (see
Sections 4.4 and 4.5) to narrow down the set of potential best approximants so we can
check (49)for the remaining candidates.
So let x =p
qbe a Markov fraction, and let x∗=p∗
q∗be a best approximant, i.e.,
suppose
Cp
q=(q∗)2·
p
q−p∗
q∗.(50)
By Lemma 2.6, this means that the Ford circle h(p∗, q∗)has the smallest signed distance
to the vertical geodesic gxamong all Ford circles except h(p, q)and h(1, 0). This leads
to the following necessary geometric condition for best approximants:
Lemma 4.8. If xis Markov fraction and x∗is a best approximant of x, then no geodesic
in the G-orbit of gxseparates gxand x∗.
(A geodesic gin the hyperbolic plane H2separates a subset A ∈H2and an ideal
point p ∈∂H2if Aand pare contained in different connected components of (H2∪
∂H2) \(g∪∂g), where ∂g is the set of the two ideal endpoints of g.)
Proof of Lemma 4.8.Suppose τ·gxseparates gxand xfor some x=p
q∈Qand τ∈G.
We have to show that xis not a best approximant of x.
Since the shortest geodesic ray from xto gxintersects τ(gx)before reaching gx,
dgx,h(p,q)>d
τ(gx),h(p,q)=dgx,τ
−1(h(p,q))
.
Since the Ford circle τ−1(h(p, q)) has a smaller signed distance to gxthan h(p, q), the
number τ−1·xapproximates xbetter than x.
Lemma 4.8 implies that x∗is a rational ideal boundary point of one of the domains
Rxor L
x
188 B. Springborn / Journal of Number Theory 263 (2024) 153–205
Fig. 16. The domains Rxand L
xbounded by xand geodesics in the G-orbit of gx.
that are bounded by the geodesic xand geodesics in the G-orbit of gx(see Fig. 16) or
one of the domains
Lx=Rx−3andR
x=L
x−3
bounded by λx=x−3and geodesics in the G-orbit of gx.
In other words, L
xis the lift of the cylinder π(Lx) adjacent to x, and R
xis the lift
of the cylinder π(Rx) adjacent to the geodesic λx. (See also Fig. 15, which shows λxand
R
x. But keep in mind that Fig. 15 does not represent the geometry of the modular torus
Mcorrectly whereas Fig. 16 does, as explained in Remark 4.1.)
We will now consider all these rational ideal boundary points that are candidates for
best approximants in turn. First, let us consider the rational ideal boundary points of
Rx, which are:
•the Markov fraction x =p
qitself, which does not qualify,
•the right companions γ+
k(x), with k≥2,
•the left companions γ−
k(x +3), also with k≥2, and
•the Markov fraction x +3.
The left companions of x +3are ruled out as best approximants because γ−
k(x +3)
has the same denominator as γ+
k(x) but is further away from x.
For a right companion
a
b=γ+
kp
q,
we get using Lemmas 3.9 and 3.12:
B. Springborn / Journal of Number Theory 263 (2024) 153–205 189
b2·
p2
q2−a
b=q2u2
k
p
q−γ+
kp
q
(6)
=qu
kuk−1
(36)
=1
3(u2
k+uk−1−1)
≥1
3(u2
2+u1−1)
(7)
=3q2≥3
This means that a right companion γ+
k(p
q) approximates the Markov fraction
p
qworse
than the Markov fractions
p1
q1and p3
q3of the centered rational Markov triple p1
q1,
p2
q2,
p3
q3
with
p2
q2=p
q.
One ideal boundary point of Rxis left to check: The Markov fraction x +3, for which
we get 12x −(x +3)
=3.
Thus we conclude that the ideal boundary points of Rxare not best approximants of
x. Similarly, the ideal boundary points of Lxare not best approximants of x.
Remark 4.9. Note that the first companion γ+
2(x)and the Markov fraction x +3ap-
proximate the Markov fraction xexactly equally badly. In general, the (k−1)st right
companion γ+
k(x)and the (k−2)nd left companion γk−1(x + 3) approximate xwith the
same approximation quality. This due to a geometric symmetry.
Now let ybe a rational ideal boundary point of L
x. Then the vertical geodesic gy
projects to a simple geodesic π(gy)in Mwith both ends in the cusp. Furthermore, π(gy)
does not intersect of the geodesic π(gx). (To see this, note that gydoes not intersect any
of the geodesics [T]k·gyfor k∈Z \{0}, nor any geodesics in the G-orbit of gx.)
This implies, first, that yis also Markov fraction. Moreover, since π(gx)and π(gy)do
not intersect, there are exactly two ideal triangulations of Mcontaining π(gx)and π(gy)
as edges. Since x <y<x +3, we can deduce that there is a unique rational Markov
triple
m=p
1
q
1
,p2
q2
,p
3
q
3with x=p2
q2
,y=p
3
q
3
.
Conversely, if mis a rational Markov triple with x =p2
q2, then
p
3
q
3is an ideal boundary
point of L
x.
For the ideal boundary point yof L
x, we get the approximation quality
(q
3)2·
p2
q2−p
3
q
3=q
3q
1
q2
,
and if mis not the unique centered rational Markov triple m, then
q
3q
1
q2
>q3q1
q2
.
190 B. Springborn / Journal of Number Theory 263 (2024) 153–205
This shows that among all ideal boundary points of L
x, the third element
p3
q3of the
centered rational Markov triple mwith x =p2
q2approximates xbest, and all other ideal
boundary points of L
xare worse.
In the same way, one sees that among all ideal boundary points of R
x, the first element
p1
q1of the centered rational Markov triple mwith x =p2
q2approximates xbest, and all
other ideal boundary points of R
xare worse.
This completes the proof of Theorem 3.6 (ii).
Remark 4.10. While many constructions and arguments in this paper generalize to ar-
bitrary hyperbolic tori with one cusp, Theorem 3.6 (ii) really depends on the geometry
of the modular torus M. For example, in the torus shown in Fig. 15, the horocycles at
γ±
2(x)are closer to gxthan the horocycles at the ideal boundary points of L
xand R
x.
4.7. Proof of Theorem 3.14 (best approximants of companions)
From the definition (6), we get immediately
q2γ±
kp
q−p
q=quk−1
uk
.
To see that also
(qu
k−1)2γ±
kp
q−γ±
k−1p
q=quk−1
uk
,
note that
γ±
kp
q−γ±
k−1p
q=1
qu
kuk−1
(51)
because
uk−1
qu
k−uk−2
qu
k−1
=1
qu
kuk−1u2
k−1−ukuk−2
and
u2
k−1−ukuk−2
(7)
=u2
k−1−(3 qu
k−1−uk−2)uk−2
=u2
k−1−3qu
k−1uk−2+u2
k−2
(36)
=1.
This shows the right equality in (37).
B. Springborn / Journal of Number Theory 263 (2024) 153–205 191
Remark 4.11. Alternatively, we could avoid these calculations here and infer from a
geometric symmetry that
p
qand γ±
k−1p
qboth approximate γ±
kp
qwith the same quality.
But we will need equation (51)anyway for the proof of the classification Theorem 2.2
(see Section 4.9).
To see the inequality in (38), note that
qu
k−1
(7)
=1
3(uk+uk−2).
So for k≥2,
quk−1
uk
=1
31+uk−2
uk≥1
3,
with equality only for k=2.
It remains to show that
quk−1
uk
<b
2γ±
kp
q−a
b(52)
for every rational number
a
b, except
p
q, γ±
k−1p
q, and γ±
kp
q. The general strategy is
the same as in the previous section. We will use Lemma 4.14 to eliminate all but a
manageable subset of candidates
a
bthat we have to check. However, instead of G-orbits
as in Lemma 4.8, we will consider orbits with respect to a larger group
G, of which Gis
a subgroup of index 2:
Definition 4.12. Let
G⊂PSL2(Z)be the group generated by the group Gof deck
transformations of Mand any lift of the hyperelliptic involution, e.g., z→ z+3.
Remark 4.13. In the previous section, it was sufficient to consider the G-orbits of gx,
because the hyperelliptic involution only reverses the orientation of simple geodesics in
Mwith both ends in the cups. Hence, the G-orbit and the
G-orbit of gxare equal if we
disregard the orientation of geodesics.
Lemma 4.14. If yis a companion of a Markov fraction and y∗is a best approximant of y,
then no geodesic in the
G-orbit of gyseparates gyand y∗.
Lemma 4.14 can be proved in the same way as the analogous Lemma 4.8 for Markov
fractions. Next, we will use it to prove a second geometric lemma:
Lemma 4.15. If yis a companion of a Markov fraction x, and y∗is a best approximant
of y, then no geodesic in the G-orbit of gxseparates gyand y∗.
192 B. Springborn / Journal of Number Theory 263 (2024) 153–205
Proof. Let us assume that yis a right companion of x, i.e.,
y=γ+
k(x),k≥2.(53)
The argument for left companions is analogous. The proof has two parts.
Part 1. We will first prove the following statement:
For all real numbers y<xthere is a geodesic in the
G-orbit of gythat separates gx
and y.
To this end, we consider two cases separately:
•y∈(−∞, y−3). Let τ(z) =z−3. Then τ∈
G, and because y−3 <x, the vertical
geodesic τ·gyseparates gxand y.
•y∈(γ+
k−1(x) −3, x). With [T] ∈Gas in Section 4.5 and τas above we have
[T]−1·y=γ+
k−1(x)and[T]−1·∞=x+3.
The geodesic τ◦[T]−1·gytherefore has ideal endpoints γ+
k−1(x) −3and x. So it
separates gyand y.
Since γ+
k−1(x) <y, the two intervals cover (−∞, x), and hence we have shown the
statement for all y<x.
Part 2. Now to prove the lemma, suppose there is a σ∈Gsuch that σ·gxseparates gy
and y. We have to show that yis not a best approximant of y.
To see this, let ˆσ∈
Gbe the element with
ˆσ·gx=σ·gxand y∈ˆσ·R<x .
(Either ˆσ=σ, or σ−1ˆσis the lift of the hyperelliptic involution on Mthat maps the
simple geodesic gxwith both ends in the cusp to itself, but reversing its orientation.)
Then ˆσ−1·y<x, so by (1) there is a geodesic g1∈
G·gythat separates gxand ˆσ−1·y.
Hence ˆσ·g1separates ˆσ·gxand y. Since ˆσ·gxseparates gyand yby assumption, the
geodesic ˆσ·g1separates gyand y. By Lemma 4.14, yis not a best approximant of y.
To prove the theorem, let us consider a right companion yas in (53). The arguments
for a left companion are completely analogous.
First, Lemma 4.15 implies that the best approximants of yare among the rational
ideal boundary points of the domains Rxand L
x(see Fig. 16).
Let us first consider the rational ideal boundary points of Rx, which are
γ+
(x)=[T]·∞for ∈Z
B. Springborn / Journal of Number Theory 263 (2024) 153–205 193
(see Section 4.5). Of these, y=γ+
k(x)and ∞=γ+
0(x)are by definition not best approx-
imants of y.
Next, we will use Lemma 4.14 to show that γ+
(x)is not a best approximant if k<
or <0. To see this, note first that the ideal endpoints of the geodesic [T]m·gyare
γ+
m(x)andγ+
m+k(x).
Hence, if
k≤m<<m+k
or
m<<m+k≤0
then [T]m·gyseparates gyand γ+
(x). Thus we have eliminated all rational ideal boundary
points of Rxas best approximants except
x=γ+
1(x),γ
+
2(x), ... γ+
k−1(x).(54)
To eliminate all rational ideal boundary points of L
x, let ι ∈
Gbe the lift of the
hyperelliptic involution on Mthat maps the geodesic xto itself, only reversing its
orientation. Then the rational ideal boundary points of L
xare the numbers
ι·γ+
(x)=ι◦[T]·∞for ∈Z
Thus, if
m<<m+k
then the geodesic ι ◦[T]m◦gyseparates gyand ι ◦γ+
(x). By Lemma 4.14, this excludes
all rational ideal boundary points of L
xas best companions of y.
Now the only candidates for best approximant of yare the numbers (54), and we
know that xand γ+
k−1(x)have the same approximation quality. To prove that they are
indeed the best approximants, it remains to show (52)for
a
b=γ+
(x)with1<<k−1.
If k=2, there is nothing to show. For k>2, we will take the geometric point of view
to simplify the calculations.
We have to show that among the Ford circles at the rational numbers (54), the minimal
signed distance to the vertical geodesic gyis attained precisely for the first and the last
(see Lemma 2.6). Note that the Ford circles at γ+
(x)with ∈Zare equally spaced
translates along the translation axis x. In particular, all these horocycles have the same
194 B. Springborn / Journal of Number Theory 263 (2024) 153–205
0e−t0et0
et1
i
het
2,e−t
2
d
Fig. 17. The signed distance dbetween the geodesic with ideal endpoints e±t0and the horocycle h(et
2, e−t
2).
signed distance to x. Moreover, 1 ≤ ≤k−1if and only if gyseparates xand the
center γ+
(x).
To simplify the calculations, we apply a hyperbolic isometry that maps the axis xto
the upward oriented vertical geodesic g0, and the geodesic gyto the geodesic with ideal
endpoints e−t0and et0for some positive real t0(see Fig. 17). Then the images of the Ford
circles at γ+
(x)are among the 1-parameter family of translates of some horocycle h(a, a)
along the axis g0. To simplify the calculations further, we will consider the translates of
h(1, 1) instead. This scales the horocycles evenly, so all distances change by the same
additive constant. That is, we consider the family of horocycles h(p(t), q(t)) with
p(t)
q(t)=et
20
0e−t
21
1=et
2
e−t
2
(see Section 2.6 and [13, p. 665], [33, Ch. 5]). Among these, the scaled images of the Ford
circles at γ+
(x)are the horocycles h(et
2, e−t
2)with
t=t0
k(2−k).(55)
(To see this, note that for integer values of between 0and k, the t-values are evenly
spaced reals between −t0and t0.)
The signed distance between the horocycle h(et
2, e−t
2)and the geodesic with ideal
endpoints e±t0is
d=log
fet
2,e
−t
2,
where fis a quadratic form on R2with determinant −1that is zero for
p
q=e±t0(see
[33, Ch. 10]). This determines fup to sign and we may choose
B. Springborn / Journal of Number Theory 263 (2024) 153–205 195
f(p, q)= 1
sinh t0p2−2(cosht0)pq +q2).
For t ∈(−t0, t0)we obtain
ed=2
sinh t0cosh t0−cosh t.(56)
This is a concave even function of twith limit zero as t →±t0. Hence, for tas in (55)
with ∈Zand 1 ≤ ≤k−1, the signed distance dattains its minimum precisely for
=1and =k−1. This completes the proof of Theorem 3.14.
4.8. Empty intervals around Markov fractions
This section is about the largest intervals Ixaround each Markov fraction xthat
contain no other Markov fractions (see Section 2.5). At the same time, Ixis the smallest
closed interval that contains all companions of x. We will prove Lemmas 4.16 and 4.18,
which will be used in the following section to complete the proof of the classification
Theorem 2.2.
From the geometric point of view, Lemmas 4.16 and 4.18 are well known, and even
in greater generality: for arbitrary hyperbolic tori with one cusp. This is the origin of
McShane’s remarkable identity for the lengths of simple closed geodesics [21](see also
Remark 4.19), which has been reproved [5,15,30]and generalized [2,22,23]in various
ways. Nevertheless, it seems worthwhile to provide independent proofs in the current
context.
Lemma 4.16. For every Markov fraction x =p
q, the interval Ixdefined by equation (11)
is the smallest closed interval that contains all companions of x. It contains no Markov
fractions other than x, nor any of their companions. But the irrational endpoints of Ix
are accumulation points of the set of Markov fractions.
Proof of Lemma 4.16.By equations (40)and(42), the right endpoint ξ+of Ixis the
limit of the increasing sequence (γ+
k(x))k≥2of right companions of x. Analogously or by
symmetry, the left endpoint is the limit of the decreasing sequence of left companions.
Thus no smaller closed interval contains all companions.
The rational ideal boundary points of the domain L
xare Markov numbers, and they
accumulate at the right endpoint of Ix(see Section 4.6 and Figs. 15 and 16). Simi-
larly, the rational ideal boundary points of R
xare Markov numbers accumulating at the
left endpoint of Ix. As accumulations points of Markov numbers, the endpoints of Ix
are irrational (see Section 2.2; in fact, it is well known that the endpoints are Markov
irrationals).
For every interior point yof Ixexcept x, the geodesic π(gy)in Mhas self-intersections.
To see this if y>x, note that gyintersects [T] ·gy(see Section 4.5). The same holds for
196 B. Springborn / Journal of Number Theory 263 (2024) 153–205
y<xby an analogous argument or by symmetry. Since the endpoints of Ixare not even
rational, this implies that xis the only Markov fraction in Ix.
In particular, this means that no other Markov fraction is between xand one of its
companions. Since this is true for all Markov fractions, and they accumulate at the
endpoints of Ix, this implies that Ixdoes not contain any companions of other Markov
fractions than x.
Corollary 4.17. For different Markov fractions x =x, the interiors
˚
Ixand
˚
Ixare dis-
joint.
Lemma 4.18. As xranges over the set of Markov fractions, the intervals Ixcover the set
Qof rational numbers.
Proof. Since the intervals obviously cover the set of Markov fractions, let us assume that
y∈Qis not a Markov fraction. We have to find a Markov fraction xfor which y∈Ix.
To this end, define the infinite sequence of centered Markov triples
mk=pk,1
qk,1
,pk,2
qk,2
,pk,3
qk,3
with
pk,1
qk,1
<y<pk,3
qk,3
(57)
recursively by
•m0=y,y+1
2,y+1, where ydenotes the largest integer not larger
than y, and
•for k>0, the triple mkis the child satisfying (57)among the two children of mk−1
in the tree of centered rational Markov triples (see Lemma 3.8).
First consider the following two cases:
1. The triple mkis the left child of mk−1for all but finitely many k.
That is, the sequence of Markov fractions
pk,1
qk,1eventually becomes constant, say
pk,1
qk,1
=xfor k>k
0,
while
pk,2
qk,2and
pk,3
qk,3are consecutive ideal boundary points of L
xwith
lim pk,2
qk,2
= lim pk,3
qk,3
=ξ+
(see Section 4.6 and Figs. 15 and 16). Hence y∈Ix
B. Springborn / Journal of Number Theory 263 (2024) 153–205 197
2. The triple mkis the right child of mk−1for all but finitely many k.
In this case,
pk,3
qk,3
=xfor k>k
0,
and like in case 1 we see that y∈Ix.
We will complete the proof of the lemma by showing that the third case does not occur:
3. Infinitely many of the triples mkare left children of mk−1and infinitely many are
right children.
For such a sequence of rational Markov triples, we will show that
lim pk,3
qk,3−pk,1
qk,1=0.(58)
Then condition (57)would imply
lim pk,1
qk,1
=y= lim pk,3
qk,3
.
This would contradict ybeing rational, because accumulation points of the set of Markov
fractions are irrational (see Section 2.2).
It remains to show (58)in case 3. To this end, note that for any rational Markov triple
p1
q1,
p2
q2,
p3
q3we have
p3
q3−p1
q1
=p3
q3−p2
q2+p2
q2−p1
q1
(18)
=
(19)
q1
q2q3
+q3
q1q2
(16)
=3−q2
q3q1
,
so it is enough to show
lim qk,2
qk,3qk,1
=3.(59)
To see this, we will show that for any centered rational Markov triple p1
q1,
p2
q2,
p3
q3we
have
q2
q3q1≥3
1+ 1
q2
1+1
q2
3
.(60)
Since both sequences (qk,1)and (qk,3)are unbounded in case 3, equation (59) follows.
198 B. Springborn / Journal of Number Theory 263 (2024) 153–205
Finally, to prove the estimate (60)for a centered rational Markov triple, note that (26)
implies
q2
q3q1≥max q1
q2q3
,q3
q1q2,
and with equation (16),
q2
q3q1≥1,
Hence
q2
q1≥q3and therefore
q1
q2q3
=q2
q3q1·q2
1
q2
2≤q2
q3q1·1
q2
3
.
In the same way we get
q2
q3≥q1and
q3
q1q2≤q2
q3q1·1
q2
1
.
Now equation (16) yields
3≤q2
q3q11
q2
3
+1+ 1
q2
1
and hence (60). This completes the proof of Lemma 4.18.
Remark 4.19. McShane’s identity [21]says that
γ
1
1+e|γ|=1
2,(61)
where the sum is taken over all simple closed geodesics in some hyperbolic torus with
one cusp, and |γ|denotes the length of the geodesic. For the modular torus M, this
identity can be derived as follows. The unoriented closed geodesics in Mare in one-to-
one correspondence with the (mod 3)-equivalence classes of Markov fractions. The length
of the closed geodesic ηxand the length of the interval Ixare related by
1
1+e|γ|=1
6|Ix|(62)
(see Remark 4.6). On the other hand, Corollary 4.17 and Lemma 4.18 imply
2
x|Ix|= 6 (63)
which yields McShane’s identity.
B. Springborn / Journal of Number Theory 263 (2024) 153–205 199
4.9. Proof of Theorem 2.2 (classification)
Since we have already proved Theorems 3.6 and 3.14 in the two previous sections,
we only have to show that C(y) <1
3if yis a rational number that is neither a Markov
fraction nor a companion of a Markov fraction.
By Lemma 4.18 there is a Markov fraction xsuch that y∈Ix. Since we assume that
yis not a Markov fraction or a companion, Lemma 4.16 implies that ylies between two
companions of xor between xand one of its first companions. In other words, there is
an integer k≥1such that
γ−
k+1(x)<y<γ
−
k(x)orγ+
k(x)<y<γ
+
k+1(x).
We will show that there is not enough space between the Ford circles at γ±
k(x)and
γ±
k+1(x)such that a vertical geodesic could pass between them and stay at least a signed
distance of log 2
3away from both. With Corollary 2.7, this shows that C(y) <1
3.
We already know that
γ±
k(x)−γ±
k+1(x)=1
qu
kuk+1
,(64)
and since the denominator a companion γ±
k(x)is quk, the euclidean radius of its Ford
circle is
1
2q2u2
k
(see equation (51), Lemma 3.9, and Section 2.6). A necessary condition for C(y) ≥1
3is
that the vertical geodesic gyfits between the Ford circles at γ±
k(x)and γ±
k+1(x), scaled
by
2
3in the euclidean metric (see Corollary 2.7 and Fig. 1). Therefore, to prove that
C(y) <1
3, it suffices to show that
γ±
k+1(x)−γ±
k(x)<1
3q2u2
k+1
+1
3q2u2
k
.(65)
And indeed we have
1
3q2u2
k+1
+1
3q2u2
k
=u2
k+u2
k+1
3q2u2
ku2
k+1
(36)
=3qu
kuk+1 +1
3q2u2
ku2
k+1
=1
qu
kuk+1
+1
3q2u2
ku2
k+1
(64)
>γ±
k(x)−γ±
k+1(x).
200 B. Springborn / Journal of Number Theory 263 (2024) 153–205
5. Coda: triangle paths
5.1. Triangle paths for Markov fractions
The Markov fraction μn
mdefined in Section 2.3 can be computed from its label n
mby
a Fibonacci-like recursion with Farey addition along triangle paths (see Fig. 18). This
construction, which is described more precisely in the following theorem, extends Propp’s
snake graph construction for Markov numbers [27].
Theorem 5.1 (Triangle paths for Markov fractions). Let m ≥0and n ≥0be coprime
integers, let zm,n =m +n ωwith ω=1
2(1 +i√3), and let αbe the oriented straight line
segment from 0to zm,n. Let τ0=[0, 1, ω], and unless (m, n)equals (1, 0) or (0, 1), let
τ1=[1,ω,1+ω], ..., τ
N=[zm,n −1,z
m,n −ω, zm,n]
be the finite sequence of triangles in the Eisenstein lattice Z +ωZthat αintersects after
τ0. Recursively label the vertices of these triangles as follows:
•Label the vertices 0, 1, and ωof τ0with
1
0,
0
1, and
1
1, respectively.
•If the vertices of τ0, ..., τj−1are already labeled, label the remaining vertex of triangle
τjwith
p1+p2
q1+q2, where
p1
q1and
p2
q2are the labels at the vertices shared with triangle τj−1.
Then the label at zm,n is the Markov fraction μn
m.
The companions of a Markov fraction can be obtained by a similar construction, which
will be described in the following section. Then, in Section 5.3, both constructions will
be derived by following geodesics in the Farey triangulation.
Remark 5.2 (Triangle paths and hyperbolic geodesics). In Fig. 18, the point 0 (labeled 1
0)
is connected to the points zm,n (labeled with the Markov fractions μn
m) not by straight
line segments as described in Theorem 5.1, but by polygonal curves that are determined
as follows. Define a hyperbolic metric on each triangle strip by equipping each trian-
gle with the hyperbolic Beltrami–Klein metric defined by its circumcircle [3, Ch. 5.1].
In this metric, the polygonal curves shown in Figs. 18 and 19 are hyperbolic geodesics
connecting their ideal endpoints. We use this construction because it works equally well
for companions (see Fig. 19), whereas the straight line segments would have vertices in
their interior. More importantly, Markov fractions and their companions correspond to
certain geodesics in the modular torus with both ends in the cusp, and the polygonal
curves shown in Figs. 18 and 19 are natural representations of these geodesics in the fol-
lowing sense: The canonical ideal triangulation of the modular torus induces a euclidean
metric on it [12], and if one uses this euclidean metric to lift the hyperbolic geodesics to
the euclidean plane, one obtains the polygonal curves shown in the figures.
B. Springborn / Journal of Number Theory 263 (2024) 153–205 201
Fig. 18. Triangle paths of the Markov fractions μn
mwith 1 ≤n<m≤5.
5.2. Triangle paths for companions
The left and right companions of a Markov fraction can also be found by a Fibonacci-
like recursion along triangle paths (see Fig. 19). Roughly speaking, while the Markov
fractions are obtained from triangle paths of primitive lattice vectors, the companions
202 B. Springborn / Journal of Number Theory 263 (2024) 153–205
Fig. 19. (a) Triangle paths for μ1
2=2
5and the first and second right companions γ+
2(2
5) =31
75 and γ+
3(2
5) =
463
1120 . (b) Same for the first and second left companions γ−
2(2
5) =29
75 and γ−
3(2
5) =433
1120 .
are obtained from triangle paths of imprimitive lattice vectors. To avoid lattice points in
their interior, the paths are bent slightly to the left or to the right, which leads to right
or left companions, respectively. More precisely:
Theorem 5.3 (Triangle paths for companions). With m, n, ω, zm,n, and αas in The-
orem 5.1, let kbe an integer ≥2, so that the oriented line segment kα from 0to
kz
m,n =zkm,kn contains k−1vertices of the Eisenstein lattice in its interior. Bend
this line segment slightly to the left or to the right while keeping the endpoints fixed to
obtain an oriented curve β+or β−, respectively, that avoids lattice points in the interior.
Or, to be more precise and more definite, let β±be the curve
β±:[0,1] →C,β
±(t)=(t±it(1 −t)) kz
m,n,
where >0is chosen small enough so that there are no lattice points between the line
segment kα and β±or in the interior of β±. As in Theorem 5.1, label the vertices 0, 1,
and ωwith
1
0,
0
1, and
1
1, respectively, and label the vertices of the triangles ˜τ±
0, ..., ˜τ±
N
crossed by β±recursively, so that in each triangle of the sequence the as yet unlabeled
vertex is labeled with the Farey sum of the already labeled vertices.
Then the label at kz
m,n is γ±
kp
q, the (k−1)st left or right companion of
p
q=μn
m.
Like the analogous construction for Markov fractions (see Theorem 5.1), this is derived
by following geodesics in the Farey triangulation (see Section 5.3).
B. Springborn / Journal of Number Theory 263 (2024) 153–205 203
5.3. Proof of Theorems 5.1 and 5.3
The Epstein–Penner convex hull construction [12] provides the modular torus Mwith
a euclidean metric, with respect to which the ideal hyperbolic triangles of the canonical
triangulation are equilateral euclidean triangles. We may assume that the developing
map D:H2→Cmaps the ideal triangle ∞, 0, 1to the euclidean triangle 0, 1, ωwith
ωas in Theorem 5.1. The image of Dis then the complement C\Γof the Eisenstein
lattice Γ =Z +ωZ. The canonical projection πfactors through C\Γ, giving rise to
the projection ¯π:
H2C\Γ
M
D
π
¯π
If mand nare coprime integers, then the projection ¯πmaps the oriented line segment
from 0to zm,n =m +nω to a simple closed curve in M. There is a unique homotopic
simple geodesic in Mwith both ends in the cusp (under homotopy with ends in the cusp),
and all simple geodesics with both ends in the cusp can be obtained in this way. If mand
nare both nonnegative, the geodesic in Mlifts to a vertical geodesic gxin H2, oriented
from ∞to 0, where xis a Markov fraction between 0and 1. As the geodesic gxtraverses
finitely many triangles of the Farey triangulation, the new vertices are obtained by Farey
addition. But the developing map Dmaps the triangles of the Farey triangulation that
gytraverses to the regular euclidean triangles that the line segment from 0to zm,n
traverses. Hence, the Markov fraction xcan also be obtained by Farey addition along
the euclidean triangle strip. This proves Theorem 5.1.
To prove Theorem 5.3 in a similar fashion, note that ¯πprojects the deformed straight
line segments to curves in Mwith both ends in the cusp that cross neither the respective
simple geodesic with both ends in the cusp nor the corresponding simple closed geodesic.
Data availability
No data was used for the research described in the article.
Acknowledgment
I would like to thank Gergely Harcos, who graciously made me aware of the previous
results by Flahive [14]and Gurwood [17].
This research was funded by the Deutsche Forschungsgemeinschaft (DFG -German
Research Foundation) - Project-ID 195170736 - TRR109.
204 B. Springborn / Journal of Number Theory 263 (2024) 153–205
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