Bull. Math. Soc. Sci. Math. Roumanie
Tome 59(107) No. 2, 2016, 175–185
Parabolas infiltrating the Ford circles
by
S. Corinne Hutchinson and Alexandru Zaharescu
Abstract
We define and study a new family of parabolas in connection with Ford
circles and discover some interesting properties that they have. Using these
properties, we provide an application of such a family. This application,
which establishes an asymptotic estimate for the L2 norm of a set of parabolas, proves useful in connection with recent work done by Kunik.
Key Words: Ford circles, Parabolas, Farey fractions.
2010 Mathematics Subject Classification: Primary 11B57; Secondary 11B99.
1
Introduction
In studying the set of fractions in 1938, L.R. Ford discovered and defined a new
geometric object known as a Ford circle. Such a circle, as defined by Ford [7], is
tangent to the x-axis at a given point with rational coordinates (a/q, 0), with a/q
in lowest terms, and centered at (a/q, 1/2q 2 ). For each such rational coordinate,
a circle of such radius exists. Ford proved the following theorem in regards to
these circles:
The representative circles of two distinct fractions are either tangent or wholly
external to one another.
In the case that two circles are tangent to one another, Ford called the representative fractions adjacent. Several mathematicians, including Ford himself,
have made connections between these circles and the set of Farey fractions. For
some recent results on the distribution of Ford circles, the reader is referred to
[1], [4]. For additional properties and results regarding Farey fractions refer to
[2], [3], [5], and those references therein.
A natural continuation of study, and that which the authors will discuss in
this paper, is the discussion of the relationship between the rational numbers
and other geometric objects, and connections that might be made between these
176
S. Corinne Hutchinson and Alexandru Zaharescu
objects and Ford circles. In this spirit we define, at each rational a/q in reduced
terms, a parabola in the following way:
Pa/q
q2
:y=
2
2
a
x−
.
q
(1.1)
These parabolas, as we will see below, exhibit many interesting geometric
properties, as well as a close relationship with Ford circles. First, we fix several
families of objects which will be important in what follows. Namely, the family
of all such parabolas as defined in (1.1), the family of Ford circles, and the family
of centers of such Ford circles:
P = Pa/q : a/q ∈ Q
C = Ca/q : a/q ∈ Q
O = Oa/q : a/q ∈ Q .
In Figure 1 these three families are shown for certain rational numbers in the
interval [0, 1].
Figure 1: Some elements of the families P, C, and O.
In the following theorem, we collect some remarkable, yet easy to prove, geometric properties of these parabolas in connection with Ford circles.
Theorem 1. Given rational numbers r1 = a1 /q1 , r2 = a2 /q2 in lowest terms,
Cr1 and Cr2 of different radii, we have the following:
(i) The center Or2 lies on the parabola Pr1 if and only if the circles Cr1 and
Cr2 are tangent.
(ii) If the circles Cr1 and Cr2 are tangent, then the parabolas Pr1 and Pr2 intersect at exactly two points which are the centers of two Ford circles.
Parabolas and Ford circles
177
(iii) If Cr3 and Cr4 are such that Cr1 is tangent to both, Cr2 is tangent to both,
and Cr1 and Cr2 are disjoint, then the parabolas Pr1 and Pr2 intersect at
two centers of Ford circles, namely Or3 and Or4 .
(iv) Reciprocity: The center Or2 lies on the parabola Pr1 if and only if Or1
lies on Pr2 .
It is clear from the theorem that there is a distinct connection between the
tangency of Ford circles and the points which lie on a given parabola as defined.
It may be interesting to study the points of intersection which are not centers of
circles to see what statements might be made about these points. We first prove
the above theorem, and then present an application of this family of parabolas.
2
Proof of theorem 1
We first prove (i). Given two circles, Cr1 and Cr2 , with r1 = a1 /q1 and r2 = a2 /q2 ,
then the radii of these circles are 1/2q12 and 1/2q22 , respectively. Therefore, the
circles are tangent if and only if the distance between their centers equals the
sum 1/2q12 + 1/2q22 . This is true if and only if
a2
a1
−
q2
q1
2
+
1
1
− 2
2q22
2q1
2
=
1
1
+ 2
2q22
2q1
2
,
(2.1)
which further simplifies to
a2
− a1 = 1 .
q2
q1 q1 q2
(2.2)
Next, the center Or2 = (a2 /q2 , 1/2q22 ) lies on the parabola Pr1 if and only if
1
q2
= 1
2
2q2
2
which reduces to
a2
a1
−
q2
q1
2
a2
1
a1 = − .
q1 q2
q2
q1
,
(2.3)
(2.4)
Therefore the center Or2 lies on the parabola Pr1 if and only if the circles Cr1
and Cr2 , are tangent. This completes the proof of (i). Similarly, the center OR1
lies on the parabola Pr2 if and only if Cr1 and Cr2 are tangent. Hence the center
Or1 lies on Pr2 if and only if Or2 lies on Pr1 . This proves (iv).
For (ii), we provide a geometric proof. Refer to Figure 2 for the proof. Given
two Ford circles Cr1 and Cr2 of different radii, which are tangent to one another,
by Ford’s theory we know that there are exactly two Ford circles, Cr3 and Cr4 ,
which are tangent to Cr1 and Cr2 . By (i), we know that Pr1 and Pr2 pass through
both centers Or3 and Or4 . Since Pr1 and Pr2 cannot intersect at more than two
points, it follows that their intersection consists exactly of Or3 and Or4 . This
178
S. Corinne Hutchinson and Alexandru Zaharescu
completes the proof of (ii). The proof of (iii) follows from (i) in the same way
as the proof of (ii) (refer to Figure 3). This completes the proof of Theorem 1.
Figure 2: The parabolas Pr1 , Pr2 , centers Or3 , Or4 , and representative circles of the
rational numbers r1 , r2 , r3 , and r4 as described in Theorem 1 (ii).
Figure 3: The parabolas Pr1 , Pr2 , centers Or3 , Or4 , and representative circles of the
rational numbers r1 , r2 , r3 , and r4 as described in Theorem 1 (iii).
Parabolas and Ford circles
3
179
The set of parabolas associated to a set of Ford circles
A natural continuation of this study of parabolas is to consider the area given
under the arcs of a certain set of parabolas. We then associate a set of parabolas
with a finite set of Ford circles. To proceed, we fix a positive integer Q. Then we
consider the finitely many Ford circles with centers on or above the horizontal
line y = 1/2Q2 . Let FQ be the ordered set of Farey fractions of order Q and let
N = N (Q) be its cardinality. One knows that
N (Q) =
3 2
Q + O(Q log Q).
π2
We now define a real valued function FQ on the interval [0, 1] as follows. Let
a/q and a0 /q 0 be consecutive fractions in FQ . From basic properties of Farey
fractions we know that the fraction θ = (a + a0 )/(q + q 0 ) does not belong to FQ ,
and it is the first fraction that would be inserted between a/q and a0 /q 0 if one
would allow Q to increase. By Ford’s theory, the circle Cθ is tangent to both
Ca/q and Ca0 /q0 . By its definition, Cθ is also tangent to the x-axis at the point
Tθ = (θ, 0). We define FQ on the interval [a/q, a0 /q 0 ] in such a way that its graph
over this interval coincides with the arc of the parabola Pθ lying above the same
interval [a/q, a0 /q 0 ]. We know from Theorem 1 (i) that the end points of this arc
are exactly the centers of the Ford circles Ca/q and Ca0 /q0 . This shows that the
function FQ is well defined and continuous on the entire interval [0, 1].
On each subinterval of the form [a/q, a0 /q 0 ], FQ is given explicitly by
(q + q 0 )2
FQ (t) =
2
a + a0
t−
q + q0
2
.
(3.1)
In Figure 4, we show the Ford circles and the parabolas described above in
the case Q = 5. Thus in this case the graph of FQ is a union of parabolas joining
those centers of Ford circles that lie on or above the line y = 1/(2 · 52 ) = 1/50.
Our next goal is to estimate the area under these parabolas, that is,
Z
Area(Q) =
1
FQ (t) dt .
0
For example, in the particular case shown in Figure 4, the area under the parabolas equals Area(5) = 0.0675925. In Table 1 one sees how Area(Q) changes as Q
increases.
Table 1: The size of Area(Q) and Q Area(Q) for selected values of Q.
180
S. Corinne Hutchinson and Alexandru Zaharescu
Figure 4: The set of Ford circles and parabolas with Q = 5.
Q
2
10
100
500
1000
5000
10000
20000
30000
40000
50000
N (Q)
3
33
3045
76117
304193
7600459
30397487
121590397
273571775
486345717
759924265
Area(Q)
0.1250000000
0.0390560699
0.0044701813
0.0009078709
0.0004549847
0.0000911752
0.0000456006
0.0000228035
0.0000152031
0.0000114026
0.0000091222
Q · Area(Q)
0.2500000000
0.3905606996
0.4470181307
0.4539354502
0.4549847094
0.4558762747
0.4560057781
0.4560703836
0.4560945837
0.4561061307
0.4561127077
Since the areas appear to decay like 1/Q, in the last column we also included
the quantity Q Area(Q).
Based on the data from the table it is natural to conjecture that the sequence Q Area(Q) Q≥1 converges to a certain constant, 0.456 . . .
The following theorem shows that this is indeed true.
Theorem 2. For all positive integers Q,
ζ(2) 1
Area(Q) =
· +O
3ζ(3) Q
log Q
Q2
,
where ζ(s) denotes the Riemann zeta function.
In particular, Theorem 2 confirms the above conjecture, as
lim Q Area(Q) = ζ(2)/3ζ(3) = 0.45614 . . .
Q→∞
Parabolas and Ford circles
181
It would be interesting to further study the oscillatory behavior of the function
Q → Q Area(Q), shown in Figure 5.
Figure 5: The behavior of the function Q → Q Area(Q).
4
Proof of theorem 2
Recall that between two consecutive Farey fractions α = a/q and β = a0 /q 0 ,
FQ (t) has the form FQ (t) = A(t − θ)2 , where θ = (a + a0 )/(q + q 0 ) is the point of
tangency to the x-axis and A = (q + q 0 )2 /2. Thus
Z
β
α
β
A(t − θ)3 FQ (t) dt =
.
3
α
(4.1)
To simplify the integral, notice that
a0
a + a0
1
−
= 0
,
q0
q + q0
q (q + q 0 )
a a + a0
1
α−θ = −
=−
,
q
q + q0
q(q + q 0 )
β−θ =
since by a fundamental property of Farey fractions, a0 q − aq 0 = 1. We find that
the integral in (4.1) reduces to
Z
β
F (t) dt =
α
1
6
1
1
+
q 3 (q + q 0 ) q 03 (q + q 0 )
.
Taking into account the symmetry of the Farey sequence with respect to 1/2,
or in other words taking into account that for each pair of consecutive Farey
fractions a/q and a0 /q 0 there is another pair having the same denominators q, q 0 ,
in reverse order, namely 1 − a0 /q 0 and 1 − a/q, one sees that
Area(Q) =
1X
1
,
3
q 3 (q + q 0 )
(4.2)
182
S. Corinne Hutchinson and Alexandru Zaharescu
where the sum is over all pairs (q, q 0 ) of consecutive denominators of the sequence
of Farey fractions of order Q. From basic properties of Farey fractions we know
that this set of pairs of consecutive denominators coincides with the set of pairs
(q, q 0 ) of integer numbers for which 1 ≤ q, q 0 ≤ Q, gcd(q, q 0 ) = 1 and q + q 0 > Q.
Thus we may rewrite (4.2) as
Area(Q) =
1
3
1
1
:= S(Q) .
q 3 (q + q 0 )
3
X
0
1≤q,q ≤Q
gcd(q,q 0 )=1
q+q 0 >Q
(4.3)
1
For convenience, we introduce the function f (x, y) = x3 (x+y)
and the triangular region Ω = Ω(Q), which is defined by the inequalities 1 ≤ x, y ≤ Q and
x + y > Q. Then the sum in (4.3) can be written as
X
S(Q) =
f (x, y) .
(x,y)∈ Ω
gcd(x,y)=1
By Möbius summation,
S(Q) =
X
(x,y)∈ Ω
=
X
f (x, y)
X
µ(d)
d| gcd(x,y)
X
µ(d)
1≤d≤Q
f (x, y) .
(4.4)
(x,y)∈ Ω
d|x, d|y
We continue the evaluation in (4.4) by changing the variables x = dm and y = dn,
to obtain
X
X
S(Q) =
µ(d)
f (dm, dn)
1≤d≤Q
(m,n)∈
X µ(d)
=
d4
d≤Q
1
dΩ
X
(m,n)∈
1
dΩ
X µ(d)
1
:=
S1 (d, Q) .
m3 (m + n)
d4
(4.5)
d≤Q
The inner sum S1 (d, Q) can be written as
S1 (d, Q) =
X
1≤m≤Q/d
=
X
1≤m≤Q/d
1
m3
1
m3
X
Q/d−m<n≤Q/d
X
Q/d<r≤Q/d+m
1
m+n
1
.
r
Here we introduce a variable s to take into account the size of r near Q/d. We
Parabolas and Ford circles
183
obtain
X
S1 (d, Q) =
1≤m≤Q/d
X
=
1≤m≤Q/d
=
=
d
Q
d
Q
m
1 X
1
h i
m3 s=1 Q + s
d
1
m3
X
1≤m≤Q/d
X
1≤m≤Q/d
m
X
Q
s=1 d
1
m3
1
m2
1
n o
− Q
+s
d
m
X
sd
1+O
Q
s=1
2
d log Q/d
+O
.
Q2
The completion of the sum over all positive integers m, yields
2
∞
d X 1
d X 1
d log Q/d
S1 (d, Q) =
−
+
O
Q m=1 m2
Q
m2
Q2
m>Q/d
2
dζ(2)
d log Q/d
=
+O
.
Q
Q2
(4.6)
Inserting this estimate into (4.5), we find that
2
X µ(d) dζ(2)
d log Q/d
ζ(2) 1
log Q
S(Q) =
+O
=
· +O
.
d4
Q
Q2
ζ(3) Q
Q2
d≤Q
Hence, in view of (4.3), the proof of the theorem is completed.
5
Application to Kunik’s function
In a recent paper, Kunik [8] defines a function t 7→ Ψ∗Q (t), for each positive integer
aj−1
Q, as follows. Between any two consecutive Farey fractions in FQ , say qj−1
and
aj
∗
,
j
=
1,
.
.
.
,
N
(Q)
−
1,
the
function
Ψ
(t)
is
defined
by:
Q
qj
qj + qj−1
aj + aj−1
aj−1
aj
Ψ∗Q (t) := −
t−
, for
≤t< .
2
qj + qj−1
qj−1
qj
The function Ψ∗Q (t) is a step function, being linear between subsequent Farey
a
fractions in FQ and having a jump of height 1/qj at each Farey fraction qjj . Kunik
established a number of interesting results, including the following L2 -estimate:
Z 1
log Q
||Ψ∗Q ||22 =
Ψ∗Q (t)2 dt = O
.
Q
0
184
S. Corinne Hutchinson and Alexandru Zaharescu
We notice a connection between the above function FQ (t) and Kunik’s function Ψ∗Q (t). More precisely, we restate an improved L2 -estimate of Kunik’s function in the following theorem.
Theorem 3. For positive integers Q ≥ 1,
Z 1
1
ζ(2) 1
log Q
Ψ∗Q (t)2 dt = Area(Q) =
· +O
,
2
6ζ(3) Q
Q2
0
where ζ(s) denotes the Riemann zeta function.
The proof of this theorem is as follows. We first notice the connection between
FQ (t) and the function Ψ∗Q (t), which says that
FQ (t) = 2Ψ∗Q (t)2 ,
for all t ∈ [0, 1] and Q ≥ 1.
From here, it is straightforward to see that by Theorem 2, one may replace the
original L2 -estimate by the one given in the theorem. Thus the proof of Theorem
3 is complete.
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Packings, Int. Math. Res. Notices (2014) doi:10.1093/imrn/rnu257. 175
[2] F. Boca, C. Cobeli, A. Zaharescu, A conjecture of R. R. Hall on Farey
points, J. Reine Angew. Math no. 535 (2001), 207–236. 175
[3] F. Boca, A. Zaharescu, The correlations of Farey fractions, J. Lond.
Math Soc. no. 72 (2005), 25–39. 175
[4] S. Chaubey, A. Malik, A. Zaharescu, k-Moments of distances between
centers of Ford circles, J. Math. Anal. Appl. no. 2 (2015), 906–919. 175
[5] C. Cobeli, A. Zaharescu, The Haros-Farey sequence at two hundred
years, Acta Univ. Apulensis Math. Inform. no. 5 (2003), 1–38. 175
[6] C. Cobeli, M. Vâjâitu, A. Zaharescu, On the intervals of a third between Farey fractions, Bull. Math. Soc. Sci. Math. Roumanie no. 3 (2010),
239–250.
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Parabolas and Ford circles
185
[8] M. Kunik, A scaling property of Farey fractions, Fakultät für Mathematik,
Otto-Von-Guericke-Universität Magdeburg, Preprint Nr. 7 (2014), 1–35.
183
Received: 27.04.2015
Revised: 06.06.2015
Accepted: 25.06.2015
Department of Mathematics,
University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA
E-mail: [email protected]
Simion Stoilow Institute of Mathematics of the Romanian Academy,
P.O. Box 1-764, RO-014700 Bucharest, Romania
and
University of Illinois, Department of Mathematics,
1409 West Green Street, Urbana, IL 61801, USA
E-mail: [email protected]