New York Journal of Mathematics
New York J. Math. 21 (2015) 637–656.
Geometry of Farey–Ford polygons
Jayadev Athreya, Sneha Chaubey, Amita Malik
and Alexandru Zaharescu
Abstract. The Farey sequence is a natural exhaustion of the set of
rational numbers between 0 and 1 by finite lists. Ford Circles are a
natural family of mutually tangent circles associated to Farey fractions:
they are an important object of study in the geometry of numbers and
hyperbolic geometry. We define two sequences of polygons associated to
these objects, the Euclidean and hyperbolic Farey–Ford polygons. We
study the asymptotic behavior of these polygons by exploring various
geometric properties such as (but not limited to) areas, length and slopes
of sides, and angles between sides.
Contents
1. Introduction and main results
1.1. Farey–Ford polygons
1.2. Geometric statistics
1.3. Moments
1.4. Distributions
2. Moments for Euclidean distance
3. Moments for hyperbolic distance and angles
3.1. Hyperbolic distance
3.2. Angles
4. Dynamical methods
5. Computations of the tail behavior of geometric statistics
5.1. Euclidean distance
5.2. Hyperbolic distance
References
638
638
640
642
643
645
646
646
650
650
650
651
652
655
Received October 16, 2014.
2010 Mathematics Subject Classification. 37A17, 11B57, 37D40.
Key words and phrases. Ford circles, Farey fractions, distribution.
J.S.A partially supported by NSF CAREER grant 1351853; NSF grant DMS 1069153;
and NSF grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric structures And
Representation varieties" (the GEAR Network).".
ISSN 1076-9803/2015
637
638
ATHREYA, CHAUBEY, MALIK AND ZAHARESCU
1. Introduction and main results
Ford circles were introduced by Lester R. Ford [6] in 1938 and since then
have appeared in several different areas of mathematics. Their connection
with Farey fractions is well known. We define the notion of Farey–Ford
polygons. We study some statistics associated to both the Euclidean and
hyperbolic geometry of these objects, in particular distances between vertices, angles, slopes of edges and areas. The analysis of these quantities is
an interesting study in itself as distances, angles and areas are among the
most common notions traditionally studied in geometry. We use dynamical
methods to compute limiting distributions, and analytic number theory to
compute asymptotic behavior of moments for the hyperbolic and Euclidean
distance between the vertices.
Below, we define Farey–Ford polygons (§1.1), describe the statistics that
we study (§1.2) and state our main theorems (§1.3 and §1.4). In §3 we prove
our results on moments, in §4 we prove our distribution results, and in §5
we compute the tail behavior of the statistics we study.
1.1. Farey–Ford polygons. Recall the definition of Ford circles: The Ford
circle
1 1
p
+ i 2 = 2
Cp/q := z ∈ C : z −
q
2q
2q
is the circle in the upper-half plane tangent to the point (p/q, 0) with diameter 1/q 2 (here and in what follows we assume that p and q are relatively
prime).
0
1
11 1
65 4
1 2
3 5
1
2
3 2
5 3
3 45
4 56
1
1
P[0,1] (5)
Figure 1. Euclidean Farey–Ford Polygons
Given δ > 0 and I = [α, β] ⊆ [0, 1], we consider the subset
1
FI,δ := Cp/q : p/q ∈ I, 2 ≥ δ
2q
GEOMETRY OF FAREY–FORD POLYGONS
639
of Ford circles with centers above the horizontal line y = δ and with tangency
points in I. Writing
j
k
Q = (2δ)−1/2 ,
we have that the set of tangency points to the x-axis is
F(Q) ∩ I,
where
F(Q) := {p/q : 0 ≤ p ≤ q ≤ Q, (p, q) = 1}
denotes the Farey fractions of order Q. We write
F(Q) ∩ I := {p1 /q1 < p2 /q2 < · · · < pNI (Q) /qNI (Q) },
where NI (Q) is the cardinality of the above set (which grows quadratically
in Q, so linearly in δ −1 ). We denote the circle based at pj /qj by Cj for
1 ≤ j ≤ NI (Q), and define the Farey–Ford polygon PI (Q) of order Q by:
• connecting (0, 1/2) to (1, 1/2) by a geodesic;
• forming the “bottom” of the polygon by connecting the points
(pj /qj , 1/2qj 2 ) and (pj+1 /qj+1 , 1/2qj+1 2 )
for 1 ≤ j ≤ NI (Q) − 1 by a geodesic.
We call these polygons Euclidean (respectively hyperbolic) Farey–Ford
polygons if the geodesics connecting the vertices are Euclidean (resp. hyperbolic). Figure 1 shows the Euclidean Farey–Ford polygon P[0,1] (5).
Remark. For n ∈ N, consider the sequence of functions fn : [0, 1] → [0, 1]

 1
if x = pq , (p, q) = 1, 1 ≤ q ≤ n,
fn (x) :=
2q 2
 0
otherwise.
In other words, fn (x) is nonzero only when x is a Farey fraction of order
n, in which case fn (x) denotes the Euclidean distance from the real axis to
the vertices of Farey–Ford polygons of order n. As we let n approach infinity,
the sequence fn (x) converges and its limit is given by 21 (f (x))2 , where f (x)
is Thomae’s function
1
if x = pq , (p, q) = 1, q > 0
q
f (x) :=
0 otherwise,
This function is continuous at every irrational point in [0, 1] and discontinuous at all rational points in [0, 1].
640
ATHREYA, CHAUBEY, MALIK AND ZAHARESCU
1.2. Geometric statistics. Given a Farey–Ford polygon PI (Q), for 1 ≤
j ≤ NI (Q) − 1, let
qj
qj+1
xj := , yj :=
.
Q
Q
For 1 ≤ j ≤ NI (Q) − 1, we have 0 < xj , yj ≤ 1. And,
xj + xj+1 =
qj + qj+1
> 1,
Q
since between any two Farey fractions
Farey fraction given by
neighbours in F(Q) ∩ I.
Let
pj +pj+1
qj +qj+1
pj
qj
and
pj+1
qj+1 ,
of order Q, there is a
which is not in F(Q) ∩ I as
pj
qj
and
pj+1
qj+1
are
Ω := {(x, y) ∈ (0, 1]2 : x + y > 1},
so we have (xj , yj ) ∈ Ω for 1 ≤ j ≤ NI (Q) − 1. A geometric statistic F is
a measurable function F : Ω → R. Given a geometric statistic F, we define
the limiting distribution for Ford circles GF (t) of F (if it exists) to be the
limiting proportion of Ford circles for which F exceeds t, that is,
#{1 ≤ j ≤ NI (Q) : F (xj , yj ) ≥ t}
.
δ→0
NI (Q)
GF (t) := lim
Similarly, we define the moment of F by
(1.1)
1
MF,I,δ (F) :=
|I|
NI (Q)−1
X
|F (xj , yj )|.
j=1
We describe how various natural geometric properties of Farey–Ford polygons can be studied using this framework.
1.2.1. Euclidean distance. Given consecutive circles Cj and Cj+1 in FI,δ ,
we consider the Euclidean distance dj between their centers Oj and Oj+1 .
Note that since consecutive circles are mutually tangent, the Euclidean geodesic connecting the centers of the circles passes through the point of tangency, and has length equal to the sum of the radii, that is,
dj =
1
1
+ 2 .
2qj2 2qj+1
Therefore, we have
Q2 dj =
1
1
+ 2.
2
2xj
2yj
Thus, by letting F (xj , yj ) = 2x1j 2 + 2y1j 2 , we have expressed the Euclidean
distance (appropriately normalized) as a geometric statistic.
GEOMETRY OF FAREY–FORD POLYGONS
641
In [4], the latter three authors computed averages of first and higher moments of the Euclidean distance dj , in essence, they computed explicit formulas for
Z 2X NIX
(Q)−1
1
dkj d(2δ)−1/2 for k ≥ 1,
NI (Q) X
j=1
where X was a large real number.
1.2.2. Slope. Let tj denote the Euclidean slope of the Euclidean geodesic
joining the centers Oj and Oj+1 of two consecutive circles Cj and Cj+1 . It
is equal to
qi
qi+1
1
.
−
tj =
2 qi+1
qi
yj
1 xj
We associate the geometric statistic F (xj , yj ) =
−
to it.
2 yj
xj
1.2.3. Euclidean angles. Let θj be the angle between the Euclidean geodesic joining Oj with (pj /qj , 0) and the Euclidean geodesic joining Oj with
the point of tangency of the circles Cj and Cj+1 . In order to compute θj ,
we complete the right angled triangle with vertices Oj , Oj+1 and the point
lying on the Euclidean geodesic joining Oj and (pj /qj , 0). Then,
tan θj =

 2q2 j qj+12
if qj+1 > qj ,
qj+1 −qj
 2q2 j qj+1
2
qj −qj+1
if qj+1 < qj .
Thus, the geometric statistic associated to tan θj is given by

j yj
 2x
if yj > xj ,
yj2 −x2j
(1.2)
F (xj , yj ) = 2xj yj
 2 2 if yj < xj .
x −y
j
j
1.2.4. Euclidean area. Let Aj denote the area of the trapezium
whose
pj
sides are formed by the Euclidean geodesics joining the points
, 0 and
qj
pj+1
pj
pj+1
Oj ;
, 0 and Oj+1 ; Oj and Oj+1 ;
, 0 and
, 0 . This gives
qj+1
qj
qj+1
1
1
Aj = 3
+
3 . The geometric statistic (normalized) in this case is
4qj qj+1 4qj qj+1
F (xj , yj ) =
1
1
+
.
3
4xj yj
4xj yj3
642
ATHREYA, CHAUBEY, MALIK AND ZAHARESCU
1.2.5. Hyperbolic distance. Given consecutive circles in FI,δ we let ρj
denote the hyperbolic distance between their centers Oj and Oj+1 , where we
use the standard hyperbolic metric on the upper half plane H defined as
p
dx2 + dy 2
ds :=
.
y
A direct computation shows that
qj+1
xj
yj
yj
ρj
qj
1 xj
+
=
+
=
+
.
sinh
=
2
2qj+1
2qj
2yj
2xj
2 yj
xj
yj
1 xj
+
, we have expressed (sinh of half of) the
Letting F (xj , yj ) =
2 yj
xj
hyperbolic distance as a geometric statistic.
1.2.6. Hyperbolic angles. Let αj denote the angle between the hyperbolic geodesic parallel to the y-axis, passing through the center Oj , and the
hyperbolic geodesic joining the centers Oj and Oj+1 . This angle can be calculated by first computing the center of the hyperbolic geodesic joining Oj
and Oj+1 . Note that the center of this geodesic lies on the x-axis. Let us
denote it by (cj , 0). Now, αj is also the acute angle formed between the line
joining (cj , 0) and Oj , and the x-axis. Thus tan αj is given by
tan αj =
4qj5
3
qj+1
− 4qj qj+1 +
16qj3
.
qj+1
Consequently the geometric statistic (normalized) associated to it is
F (xj , yj ) =
4x5j
yj3
− 4xj yj +
16x3j
.
yj
1.3. Moments. The following results concern the growth of the moments
MF,I,δ for the geometric statistics expressed in terms of Euclidean distances,
hyperbolic distances and angles.
Theorem 1.1. For any real number δ > 0, and any interval I = [α, β] ⊂
[0, 1], with α, β ∈ Q and F (x, y) = 2x12 + 2y12 ,
6 2
Q log Q + DI Q2 + OI (Q log Q) ,
π2
where MF,I,δ (F) is defined as in (1.1), Q is the integer part of √12δ and DI
is a constant depending only on the interval I. Here the implied constant in
the Big O-term depends on the interval I.
MF,I,δ (F) =
Theorem 1.2. For any real number δ > 0, any interval I = [α, β] ⊂ [0, 1]
1 x
and F (x, y) = 2 y + xy ,
9
MF,I,δ (F) = 2 Q2 + OI, Q7/4+ ,
2π
GEOMETRY OF FAREY–FORD POLYGONS
643
where MF,I,δ (F) is defined as in (1.1), CI is a constant depending only on
the interval I, is any positive real number, and Q is the integer part of √12δ .
The implied constant in the Big O-term depends on the interval I and .
Theorem 1.3. For any real number δ > 0, any interval I = [α, β] ⊂ [0, 1]
and F (x, y) as in (1.2), we have,
12
MF,I,δ (F) = Q2 + OI (Q log Q) ,
π
where MF,I,δ (F) is defined as in (1.1), and Q is the integer part of √12δ . The
implied constant in the Big O-term depends on the interval I.
1.4. Distributions. The equidistribution of a certain family of measures
on Ω yields information on the distribution of individual geometric statistics
and also on the joint distribution of any finite family of shifted geometric
statistics. In particular, we can understand how all of the above statistics
correlate with each other, and can even understand how the statistics observed at shifted indices correlate with each other. First, we record the result
on equidistribution of measures.
Let
N
1 X
δ(xj ,yj )
ρQ,I :=
N
i=1
be the probability measure supported on the set
qj qj+1
(xj , yj ) =
,
:1≤j≤N .
Q Q
We have [2, Theorem 1.3] (see also [10], [11]):
Theorem 1.4. The measures ρQ,I equidistribute as Q → ∞. That is,
lim ρQ,I = m,
Q→∞
where dm = 2dxdy is the Lebesgue probability measure on Ω and the convergence is in the weak-∗ topology.
As consequences, we obtain results on individual and joint distributions
of geometric statistics.
Corollary 1. Let F : Ω → R+ be a geometric statistic and let m denote
the Lebesgue probability measure on Ω. Then for any interval I ⊂ [0, 1], the
limiting distribution GF (t) exists, and is given by
GF (t) = m F −1 (t, ∞) .
In order to obtain results on joint distributions, we will need the following
notations. Let T : Ω → Ω be the BCZ map,
1+x
T (x, y) := y, −x +
y .
y
644
ATHREYA, CHAUBEY, MALIK AND ZAHARESCU
A crucial observation, due to Boca, Cobeli and one of the authors [3] is that
T (xj , yj ) = (xj+1 , yj+1 ).
Let F = {F1 , . . . , Fk } denote a finite collection of geometric statistics, and
let n = (n1 , . . . , nk ) ∈ Zk and t = (t1 , . . . , tk ) ∈ Rk . Define GF,n (t), the
(F, n)-limiting distribution for Ford circles by the limit (if it exists),
#{1 ≤ j ≤ NI (Q) : Fi (xj+ni , yj+ni ) ≥ ti , 1 ≤ i ≤ k}
,
δ→0
NI (Q)
GF,n (t) := lim
where the subscripts are viewed modulo NI (Q). This quantity reflects the
proportion of circles with specified behavior of the statistics Fi at the indices
j + ni . Corollary 1 is in fact a special (k = 1, n = 0) case of the following.
Corollary 2. For any finite collection of geometric statistics
F = {F1 , . . . , Fk },
n = (n1 , . . . , nk ) ∈ Zk ,
t = (t1 , . . . , tk ) ∈ Rk ,
the limiting distribution GF,n (t) exists, and is given by
GF,n (t) = m (F ◦ T )−n (t1 , . . . , tk ) ,
where
−n
(F ◦ T )
(t1 , . . . , tk ) :=
k
\
(Fi ◦ T ni )−1 (ti , ∞).
i=1
1.4.1. Tails. For the geometric statistics listed in Section 1.2, one can also
explicitly compute tail behavior of the limiting distributions, that is, the
behavior of GF (t) as t → ∞. We use the notation f (x) ∼ g(x) as x → ∞ to
mean that
f (x)
= 1,
lim
x→∞ g(x)
and the notation f (x) g(x) means that there exist positive constants c1 , c2
and a constant x0 such that for all x > x0 ,
c1 |g(x)| ≤ |f (x)| ≤ c2 |g(x)|.
We have the following result for the tail behavior of limiting distributions
for various geometric statistics.
Theorem 1.5. Consider the geometric statistics associated to the Euclidean
distance, hyperbolic distance, Euclidean slope, Euclidean area and hyperbolic
angle. The limiting distribution functions associated to these have the following tail behavior:
(1) (Euclidean distance) for F (x, y) = 2x12 + 2y12 , GF (t) ∼ 1t .
(2) (Hyperbolic distance) for F (x, y) = 12 xy + xy , GF (t) ∼ 2t12 .
(3) (Euclidean slope) for F (x, y) = 12 xy − xy , GF (t) t12 .
1
(4) (Euclidean area) for F (x, y) = 14 x13 y + xy1 3 , GF (t) t2/3
.
(5) (Hyperbolic angle) for F (x, y) =
4x5
y3
− 4xy +
16x3
y ,
GF (t) 1
.
t2/3
GEOMETRY OF FAREY–FORD POLYGONS
645
2. Moments for Euclidean distance
In this section, we prove Theorem 1.1. Recall that the Euclidean distance
dj between the centers of two consecutive Ford circles Cj and Cj+1 is given
by
1
1
dj =
+
.
2
2qj
2qj+1 2
As mentioned in section 1.2.1, the geometric statistic F expressed in terms
of the Euclidean distance dj is
F (xj , yj ) = Q2 dj .
Thus, the moment of F is given by
Q2
MF,I,δ (F) =
|I|
NI (Q)−1
X
j=1
1
1
+ 2
2
2qj
2qj+1
!
.
Proof of Theorem 1.1.
(2.1)
|I|
ME,I,δ (F)
Q2
NI (Q)
X 1
1
1
+ 2− 2
2
qj
2q1
2qNI (Q)
j=2
X 1
1
1
= 2− 2
+
q2
2q1
2qN (Q)
=
1≤q≤Q
I
=
X 1
1
1
− 2
+
2
q2
2q1
2qNI (Q)
q≤Q
=
1
1
− 2
2q12 2qN
I (Q)
X
1
αq<a≤βq
(a,q)=1
X
αq<a≤βq
X
1
µ(d)
d|(a,q)
X 1 X
X
+
µ(d)
1
2
q
αq
βq
q≤Q
d|q
d
<l≤
d
1
1
= 2− 2
2q1
2qNI (Q)


n o
X 1 X
X
(β
−
α)q
βq
αq


µ(d)
−
µ(d)
−
+
q2
d
d
d
q≤Q
d|q
d|q
X φ(q) X µ(d) X {βm} − {αm} −
+
= |I|
q2
d2
m2
Q
q≤Q
d≤Q
m≤ d
1
1
− 2
2
2q1
2qNI (Q)
!
.
Note that since α, β ∈ Q, for large Q one can assume that α and β are Farey
1
1
is a constant.
fractions of order Q. Therefore, the quantity 2 − 2
2q1
2qNI (Q)
646
ATHREYA, CHAUBEY, MALIK AND ZAHARESCU
Now,
X {βm} − {αm} m2
m≤ Q
d
=
∞ X
{βm} − {αm}
m2
m=1
−
X {βm} − {αm} m2
Q
m> d
= CI + O I
d
Q
.
This gives
X µ(d) {βm} − {αm} CI
log Q
=
.
+ OI
d2
m2
ζ(2)
Q
d≤Q
Combining this in (2.1) and the fact that
X φ(q)
6
log Q
(2.2)
=
log
Q
+
A
+
O
,
q2
π2
Q
q≤Q
we obtain,
6 log Q
+ DI + O I
ME,I,δ (F) =
π2
log Q
Q
.
It only remains to prove (2.2).
X φ(n) X 1 X
X µ(d)
n
=
µ(d)
=
n2
n2
d
qd2
n≤x
n≤x
d|n
q,d
qd≤x
X µ(d) X 1 X µ(d) 1
=
=
log x − log d + A + O
2
2
d
q
d
x
d≤x
d≤x
q≤x/d
!
!
∞
∞
X
X
µ(d) X µ(d)
µ(d) log d X µ(d) log d
= log x
−
−
−
d2
d2
d2
d2
d=1
d>x
d>x
d=1
!
∞
X
µ(d) X µ(d)
1
+A
−
+O
2
2
d
d
x
d=1
d>x
log x
log x
+B+O
.
=
ζ(2)
x
This completes the proof of Theorem 1.1.
3. Moments for hyperbolic distance and angles
3.1. Hyperbolic distance. In this section, we prove Theorem 1.2. Recall
that ρj denotes the hyperbolic distance between consecutive centers of Ford
circles Oj and Oj+1 , and
ρj
qj
qj+1
(3.1)
sinh
=
+
.
2
2qj+1
2qj
GEOMETRY OF FAREY–FORD POLYGONS
647
ρ
In order to compute the total sum of the distances sinh 2j ,
X
ρj
q
q0
1 X
1
(3.2)
+
,
MH,I,δ (F) =
sinh
=
|I|
2
|I| 0
2q 0 2q
j
q,q neighbours
in FI (Q)
we use the following key analytic lemma.
Lemma 3.1. [3, Lemma 2.3] Suppose that 0 < a < b are two real numbers,
and q is a positive integer. Let f be a piecewise C 1 function on [a, b]. Then
Z b
Z b
X φ(k)
6
0
|f (x)| dx
.
f (x) dx + OI log b ||f ||∞ +
f (k) = 2
k
π a
a
a<k≤b
We also use the Abel summation formula,
(3.3)
X
Z
a(n)f (n) = A(y)f (y) − A(x)f (x) −
y
A(t)f 0 (t) dt,
x
x<n≤y
where a : N → C is an arithmetic function, 0 < x < y are real numbers,
f : [x, y] → C is a function with continuous derivative on [x, y], and
X
A(t) :=
a(n).
1≤n≤t
Proof of Theorem 1.2. Let q¯0 denote the unique multiplicative inverse of
q 0 modulo q in the interval [1, q]. Define
(
1 if (q, q 0 ) = 1 and q¯0 ∈ [qα, qβ],
0
aq (q ) =
0 otherwise.
Using Weil type estimates ([5], [9], [14]) for Kloosterman sums, the partial
sum of aq (q 0 ) was estimated in [3]. More precisely, by [3, Lemma 1.7], for
any > 0,
X
φ(q)
(3.4)
Aq (t) :=
aq (q 0 ) =
(t − 1)|I| + OI, q 1/2+ .
q
0
1≤q ≤t
In order to estimate the sum in (3.2), we write it as
X
X
q
q0
(3.5) 2|I|MH,I,δ (F) =
+
+
0
q
q
0
0
q,q neighbours
in FI (Q)
q≥q 0
q,q neighbours
in FI (Q)
q<q 0
q
q0
+
q0
q
.
Note that it suffices to estimate the first sum above because of the symmetry
in q and q 0 . And for the first sum, one has:
648
ATHREYA, CHAUBEY, MALIK AND ZAHARESCU
X
(3.6)
q,q 0
neighbours
in FI (Q)
q≥q 0
q
q0
+
q0
q
X
=
0
aq (q )
Q/2<q≤Q−L
Q−q<q 0 ≤q
+
X
0
q0
q
+
q0
q
q
q0
+
q0
q
q
q0
+
q0
q
aq (q )
Q−L<q≤Q
L<q 0 ≤q
+
X
0
aq (q )
Q−L<q≤Q
Q−q<q 0 ≤L
=: S1 + S2 + S3 .
We begin with estimating the sum S1 .
X
X
q0
q
=:
S1 =
aq (q 0 ) 0 +
q
q
0
Q/2<q≤Q−L Q−q<q ≤q
X
Σ1 .
Q/2<q≤Q−L
Employing (3.4), (3.3) and the fact that q > Q − q, for the inner sum Σ1 in
S1 , we have
q q
Q−q
q
(3.7) Σ1 = Aq (q)
+
− Aq (Q − q)
+
q q
q
Q−q
Z q
q
1
− 2 dt
−
Aq (t)
q t
Q−q
φ(q)
1/2+
=2
(q − 1)|I| + OI, q
q
Q − q
φ(q)
q
1/2+
−
(Q − q − 1)|I| + OI, q
+
q
q
Q−q
Z q 1
φ(q)
q
−
(t − 1)|I| + OI, q 1/2+
− 2 dt
q
q t
Q−q
φ(q)
Q−q
q
= |I|
2(q − 1) − (Q − q − 1)
+
q
q
Q−q
!
!
Z q
q 1
q
1/2+
−
(t − 1)
−
dt + OI, q
q t2
Q−q
Q−q
φ(q)
Q−q
q
Q2 Q
= |I|
2(q − 1) − (Q − q − 1)
+
+
−
q
q
Q−q
2q
q
!
q 3/2+
q
−
− q log(Q − q) + q log q − Q + 3 + OI,
Q−q
Q−q
!
Q2
q 3/2+
=−
− q log(Q − q) + q log q + Q + OI,
.
2q
Q−q
GEOMETRY OF FAREY–FORD POLYGONS
649
Therefore,
S1 = |I|
X
Q
<q≤Q−L
2
φ(q)
g(q) + OI,
q
Q5/2+
L
!
,
2
where g(q) := − Q2q − q log(Q − q) + q log q + Q. Also, from (3.7) and Lemma 3.1, we obtain
!
Z Q−L
Z
6|I| Q−L
0
(3.8) S1 = 2
|g (x)| dx)
g(x) dx + OI, log Q(||g||∞ +
π
Q/2
Q/2
!
Q5/2+
+ OI,
.
L
And,
Z
Q−L
(3.9)
Q/2
(3.10) ||g||∞
Q2 − (Q − L)2
L
3(Q − L)Q 3Q2
g(q) =
log
+
−
;
2
Q−L
2
4
Z Q−L
= OI (Q log Q) =
|g 0 (x)| dx.
Q/2
Therefore, from (3.8), (3.9), (3.10) and Lemma 3.1,
L
3(Q − L)Q 3Q2
6|I| Q2 − (Q − L)2
(3.11) S1 = 2
log
+
−
π
2
Q−L
2
4
!
Q5/2+
+ OI,
+ OI (Q log Q) .
L
Next, we estimate the sum S2 in (3.6). Recall that since q 0 ≤ q, we have


0
X
X
X
X q
q
q

(3.12) S2 =
aq (q 0 ) 0 +
= OI 
q
q
q0
0
0
Q−L<q≤Q L<q ≤q
Q−L<q≤Q L<q ≤q


X
X 1
 = OI (QL log Q) .
= OI 
q
q0
0
Q−L<q≤Q
L<q ≤q
Lastly,
(3.13)
S3 =
X
X
Q−L<q≤Q
Q−q<q 0 ≤L
aq (q )

= OI 
X
0
X
1≤q 0 ≤L Q−q 0 <q≤Q
q0
q
+
q0
q

q
= OI (QL) ,
q0
650
ATHREYA, CHAUBEY, MALIK AND ZAHARESCU
Setting L = Q3/4 , and from (3.11), (3.12), (3.13), and (3.6), we obtain the
first moment for hyperbolic distances as
9
MH,I,δ (F) = 2 Q2 + OI, Q7/4+ .
2π
3.2. Angles. First we consider the case when qj+1 is smaller than qj . Let
θj denote the angle for the circle Cj . Then, the sum of the angles for both
circles is given by θj + π2 − θj+1 + π2 = π. Similarly, in the case qj < qj+1 ,
the sum of the angles is π. This gives
NI (Q)
2π
12
1 X
(θj + θj+1 ) =
NI (Q) = Q2 + OI (log Q) .
Mθ,I,δ (F) =
|I|
|I|
π
j=1
4. Dynamical methods
In this section, we prove Corollaries 1 and 2. As we stated before, they
are both consequences of the equidistribution of a certain family of measures
on the region
Ω = {(x, y) ∈ (0, 1]2 : x + y > 1}.
To see this, we write (with notation as in §1.4)
#{1 ≤ j ≤ NI (Q) : F (xj , yj ) ≥ t}
= ρQ,I F −1 (t, ∞)
NI (Q)
#{1 ≤ j ≤ NI (Q) : Fi (xj+ni , yj+ni ) ≥ ti , 1 ≤ i ≤ k}
NI (Q)
= ρQ,I (F ◦ T )−n (t1 , . . . , tk )
Now the results follow from applying Theorem 1.4 to the above expressions.
Remark (Shrinking Intervals). Versions of Corollaries 1 and 2 for shrinking
intervals IQ = [αQ , βQ ] where the difference βQ − αQ is permitted to tend
to zero with Q can be obtained by replacing Theorem 1.4 with appropriate discretizations of results of Hejhal [8] and Strömbergsson [13], see [2,
Remark 1].
5. Computations of the tail behavior of geometric statistics
In this section, we compute the tails of our geometric statistics (§1.4.1).
GEOMETRY OF FAREY–FORD POLYGONS
651
1
1
5.1. Euclidean distance. For F (x, y) = 2 + 2 , by Corollary 1,
2x
2y
1
1
−1
.
GF (t) = m(F (t, ∞)) = m
(x, y) ∈ Ω, 2 + 2 ≥ 2t
x
y
Define
1
1
Bt := (x, y) ∈ Ω, 2 + 2 ≥ 2t .
x
y
Therefore,
Z Z
Z
dx dy.
dm = 2
GF (t) =
Bt
Bt
1
1
+ 2 ≥ 2t automatically holds true.
2
x √y
The region bounded by lines 0 < y < 1/ 2t and 1 − y < x < 1 is exactly
1
an isosceles right triangle with area . Let us assume in what follows that
4t
y > √12t . Then,
Note that if y <
√1 ,
2t
the inequality
1
x≤ q
2t −
1
y2
.
This yields


1
1 − y < x < min 1, q

2t −
Therefore, 1−y <
q 1
2t−
1
y2


1
y2
.

, which implies either α1 < y < α2 or α3 < y < α4 ,
where
q
√
t − t2 + 2t + 2t( 1 + 2t)
α1 :=
2t
t−
;
q
α2 :=
√
t2 + 2t − 2t( 1 + 2t)
2t
,
and
t+
α3 :=
q
√
t2 + 2t − 2t( 1 + 2t)
2t
t+
;
α4 :=
q
√
t2 + 2t + 2t( 1 + 2t)
2t
.
Also,
1
1
α1 < 0 < √ < √
< α2 < α3 < 1 < α4 .
2t − 1
2t
√
Hence a point (x, y) with y > 1/ 2t belongs to Bt if and only if




1
1
(5.2)
1 − y < x < min 1, q
and y ∈ √ , α2 ∪ (α3 , 1).

2t
2t − 12 
(5.1)
y
Now for the measure of the set Bt minus the isosceles right triangle already
discussed above, we consider the following two cases.
652
ATHREYA, CHAUBEY, MALIK AND ZAHARESCU
(
)
1
2t−
Case I. min 1, q
= 1, i.e., y ≤
1
y2
we have that 1 − y < x < 1 and
(
)
1
2t−
Case II. min 1, q
q 1
2t−
=
1
y2
√1
2t
√ 1 .
2t−1
<y≤
1
y2
and (5.2), we have that 1 − y < x <
Employing (5.1) and (5.2),
√ 1 .
2t−1
, i.e., y ≥
q 1
2t−
1
y2
√ 1 .
2t−1
Then, using (5.1)
√ 1
2t−1
and either
< y < α2 or
α3 < y < 1.
Combining the two cases above and adding the contribution from the
isosceles right triangle, we obtain
Z √1 Z 1
2t−1
1
GF (t) =
(5.3)
+2
dx dy
2t
√1
1−y
2t
Z
α2
+2
√ 1
2t−1
Z
r
1
2t− 12
y
Z
1
Z
r
dx dy + 2
1−y
α3
1
2t− 12
y
dx dy
1−y
r
q
√
4t − 1
1
1
1
=√
−1−
+
t t − 2 2t + 1 + 2
t 2t − 1
t
2t − 1
r
q
√
√
1
+
t − 2t + 1 − t t − 2 2t + 1 + 2
tr
q
√
√
1
t − 2t + 1 + t t − 2 2t + 1 + 2 .
−
t
This gives
1
GF (t) ∼ .
t
1 y x
5.2. Hyperbolic distance. In this case F (x, y) =
+
and
2 x y
1 y x
GF (t) = m(Ct ), where Ct := (x, y) ∈ Ω,
+
≥t .
2 x y
1
2
y x
+
x y
≥ t ⇒ either x ≥ y(t +
p
p
t2 − 1) or x ≤ y(t − t2 − 1).
Since x, y ∈ Ω, we note that either
n
o
p
1 − y < x < min 1, y(t − t2 − 1)
or
n
o
p
max 1 − y, y(t + t2 − 1) < x < 1.
Depending on the bounds of x and y, we have the following cases:
GEOMETRY OF FAREY–FORD POLYGONS
653
n
o
√
Case I. 1 − y < x < min 1, y(t − t2 − 1) . As 0 < y < 1 and
p
1
√
t − t2 − 1 =
< 1,
t + t2 + 1
√
we note that y(t − t2 − 1) < 1 and therefore
n
o
p
p
min 1, y(t − t2 − 1) = y(t − t2 − 1).
√
1
.
Also,
y(t
−
t2 − 1) < 1 implies
This implies y > t+1−√
2
t −1
−1
p
p
y < t − t2 − 1
= t + t2 − 1,
which always holds since y < 1 and t 1. And so in this case the range for
x and y is given by,
p
1
√
(5.4)
< y < 1 and 1 − y < x < y(t − t2 − 1).
t + 1 − t2 − 1
n
o
√
Case II. max 1 − y, y(t + t2 − 1) < x < 1. Here, we consider two cases.
n
o
√
Subcase I. max 1 − y, y(t + t2 − 1) = 1 − y. Then
y<
√
1
−1+1
t2
t+
and the range for x and y is given by
1
√
(5.5)
0<y<
and 1 − y < x < 1.
2
t+ t −1+1
n
o
√
√
Subcase II. When max 1 − y, y(t + t2 − 1) = y(t + t2 − 1). This implies
p
1
√
y>
and y(t + t2 − 1) < 1,
t + t2 − 1 + 1
and so we get,
p
1
1
√
√
(5.6)
<y<
and y(t + t2 − 1) < x < 1.
t + t2 − 1 + 1
(t + t2 − 1)
Combining all the above cases, from (5.4), (5.5) and (5.6), we have
Z 1
Z y(t−√t2 −1)
Z √1
Z 1
t+ t2 −1+1
dx dy + 2
dx dy
GF (t) = 2
√
t−
Z
+2
1
t2 −1+1
√1
√
t+
=
t2 −1
t+
1
t2 −1+1
1−y
Z
0
1−y
1
√
y(t+ t2 −1)
dx dy
1
1
√
√
+
2
2
(t +
− 1)(t + 1 + t − 1) (t + t − 1 + 1)2
1
√
√
+
2
(t + t − 1)(t + t2 − 1 + 1)2
√
t2
654
ATHREYA, CHAUBEY, MALIK AND ZAHARESCU
=
(t +
√
t2
2
√
.
− 1)(t + t2 − 1 + 1)
Thus,
GF (t) ∼
1.0
-2.4
0.8
-1.92
-1.44
1
.
2t2
1.0
-0.48
60
483624
0.8
-0.96
0.48
0.6
12
y
0
y
0.6
0.4
0.4
1.44
0.96
0.2
0.0
0.0
0.2
0.4
0.6
1.92
2.4
0.2
0.8
1.0
0.0
0.0
24
36
48
60
0.2
0.4
x
0.6
0.8
1.0
x
(a) Level curves for geometric statis- (b) Level curves for geometric statistic for Euclidean slope in the region tic for Euclidean area in the region Ω.
Ω.
The tails for the geometric statistics for Euclidean slope, Euclidean area
and hyperbolic angle can be computed in a similar manner.
1.0
14
0.8
28
42
0.4
56
70
84
98
y
0.6
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
x
(a) Level curves for geometric
statistic for hyperbolic angle.
Acknowledgement. The authors are grateful to the referee for many useful
suggestions and comments.
GEOMETRY OF FAREY–FORD POLYGONS
655
References
[1] Apostol, Tom M. Introduction to analytic number theory. Undergraduate Texts in
Mathematics. Springer-Verlag, New York-Heidelberg, 1976. xii+338 pp. MR0434929
(55 #7892), Zbl 0335.10001, doi: 10.1007/978-1-4757-5579-4.
[2] Athreya, Jayadev S.; Cheung, Yitwah. A Poincaré section for the horocycle
flow on the space of lattices. Int. Math. Res. Not. IMRN 2014, no. 10, 2643–2690.
MR3214280, Zbl 06369450, arXiv:1206.6597, doi: 10.1093/imrn/rnt003.
[3] Boca, Florin P.; Cobeli, Cristian; Zaharescu, Alexandru. Distribution of
lattice points visible from the origin. Comm. Math. Phys. 213 (2000), no. 2, 433–470.
MR1785463 (2001j:11094), Zbl 0989.11049, doi: 10.1007/s002200000250.
[4] Chaubey, Sneha; Malik, Amita; Zaharescu, Alexandru. k-moments of distances between centers of Ford circles. J. Math Anal. Appl. 422 (2015), no. 2, 906–
919. MR3269490, Zbl 1308.11017, doi: 10.1016/j.jmaa.2014.09.007.
[5] Estermann, T. On Kloosterman’s sum. Mathematika 8 (1961), 83–86. MR0126420
(23 #A3716), Zbl 0114.26302, doi: 10.1112/S0025579300002187.
[6] Ford, L.R. Fractions. Amer. Math. Monthly 45 (1938), no. 9, 586–601. MR1524411,
Zbl 0019.39505, doi: 10.2307/2302799.
[7] Hejhal, Dennis A. On value distribution properties of automorphic functions along
closed horocycles. XVIth Rolf Nevanlinna Colloquium (Joensuu, 1995), 39–52. de
Gruyter, Berlin, 1996. MR1427069 (97j:11024), Zbl 0867.11030.
[8] Hejhal, Dennis A. On the uniform equidistribution of long closed horocycles.
Asian J. Math. 4 (2000), no. 4, 839–853. MR1870662 (2002j:11049), Zbl 1014.11038,
doi: 10.1215/S0012-7094-04-12334-6.
[9] Hooley, C. An asymptotic formula in the theory of numbers. Proc. London Math. Soc. (3) 7 (1957), 396–413. MR0090613 (19,839c), Zbl 0079.27301,
doi: 10.1112/plms/s3-7.1.396.
[10] Kargaev, Pavel; Zhigljavsky, Anatoly. Asymptotic distribution of the distance
function to the Farey points. J. Number Theory 65 (1997), no. 1, 130–149. MR1458209
(98m:11077), Zbl 0931.11028, doi: 10.1006/jnth.1997.2110.
[11] Marklof, Jens. Fine-scale statistics for the multidimensional Farey sequence. Limit
theorems in probability, statistics and number theory, 49–57, Springer Proc. Math.
Stat., 42. Springer, Heidelberg, 2013. MR3079137, Zbl 1291.37044, arXiv:1207.0954,
doi: 10.1007/978-3-642-36068-8_3.
[12] Sarnak, Peter. Asymptotic behavior of periodic orbits of the horocycle flow and
Eisenstein series. Comm. Pure Appl. Math. 34 (1981), no. 6, 719–739. MR0634284
(83m:58060), Zbl 0501.58027, doi: 10.1002/cpa.3160340602.
[13] Strömbergsson, Andreas. On the uniform equidistribution of long closed horocycles. Duke Math. J. 123 (2004), no. 3, 507–547. MR2068968 (2005f:11105), Zbl
1060.37023, doi: 10.1215/S0012-7094-04-12334-6.
[14] Weil, André. On some exponential sums. Proc. Nat. Acad. U.S.A. 34 (1948), 204–
207. MR0027006 (10,234e), Zbl 0032.26102, doi: 10.1073/pnas.34.5.204.
656
ATHREYA, CHAUBEY, MALIK AND ZAHARESCU
(Jayadev Athreya) Department of Mathematics, University of Illinois, 1409
West Green Street, Urbana, IL 61801, USA.
jathreya@illinois.edu
(Sneha Chaubey) Department of Mathematics, University of Illinois, 1409 West
Green Street, Urbana, IL 61801, USA.
chaubey2@illinois.edu
(Amita Malik) Department of Mathematics, University of Illinois, 1409 West
Green Street, Urbana, IL 61801, USA.
amalik10@illinois.edu
(Alexandru Zaharescu) Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania.
Department of Mathematics, University of Illinois, 1409 West Green Street,
Urbana, IL 61801, USA.
zaharesc@illinois.edu
This paper is available via http://nyjm.albany.edu/j/2015/21-28.html.