Parabolas infiltrating the Ford circles

Transcription

1 Bull. Math. Soc. Sci. Math. Roumanie Tome 59(07) No. 2, 206, 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 L 2 norm of a set of parabolas, proves useful in connection with recent work done by Kunik. Key Words: Ford circles, Parabolas, Farey fractions. 200 Mathematics Subject Classification: Primary B57; Secondary B99. Introduction In studying the set of fractions in 938, 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, /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 [], [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

2 76 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 (.) 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 (.), the family of Ford circles, and the family of centers of such Ford circles: P = Pa/q : a/q C = Ca/q : a/q O = Oa/q : a/q. In Figure these three families are shown for certain rational numbers in the interval [0, ]. Figure : 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. Given rational numbers r = a /q, r2 = a2 /q2 in lowest terms, Cr and Cr2 of different radii, we have the following: (i) The center Or2 lies on the parabola Pr if and only if the circles Cr and Cr2 are tangent. (ii) If the circles Cr and Cr2 are tangent, then the parabolas Pr and Pr2 intersect at exactly two points which are the centers of two Ford circles.

3 Parabolas and Ford circles 77 (iii) If C r3 and C r4 are such that C r is tangent to both, C r2 is tangent to both, and C r and C r2 are disjoint, then the parabolas P r and P r2 intersect at two centers of Ford circles, namely O r3 and O r4. (iv) Reciprocity: The center O r2 lies on the parabola P r if and only if O r lies on P r2. 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 We first prove (i). Given two circles, C r and C r2, with r = a /q and r 2 = a 2 /q 2, then the radii of these circles are /2q 2 and /2q 2 2, respectively. Therefore, the circles are tangent if and only if the distance between their centers equals the sum /2q 2 + /2q 2 2. This is true if and only if ( a2 a ) 2 ( + q 2 q 2q2 2 ) 2 ( 2q 2 = 2q2 2 + ) 2 2q 2, (2.) which further simplifies to a 2 a q 2 =. (2.2) q q 2 q Next, the center O r2 = (a 2 /q 2, /2q 2 2) lies on the parabola P r if and only if 2q 2 2 = q2 2 ( a2 a ) 2, (2.3) q 2 q which reduces to q q 2 = a 2 a q 2. (2.4) q Therefore the center O r2 lies on the parabola P r if and only if the circles C r and C r2, are tangent. This completes the proof of (i). Similarly, the center O R lies on the parabola P r2 if and only if C r and C r2 are tangent. Hence the center O r lies on P r2 if and only if O r2 lies on P r. This proves (iv). For (ii), we provide a geometric proof. Refer to Figure 2 for the proof. Given two Ford circles C r and C r2 of different radii, which are tangent to one another, by Ford s theory we know that there are exactly two Ford circles, C r3 and C r4, which are tangent to C r and C r2. By (i), we know that P r and P r2 pass through both centers O r3 and O r4. Since P r and P r2 cannot intersect at more than two points, it follows that their intersection consists exactly of O r3 and O r4. This

4 78 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. Figure 2: The parabolas P r, P r2, centers O r3, O r4, and representative circles of the rational numbers r, r 2, r 3, and r 4 as described in Theorem (ii). Figure 3: The parabolas P r, P r2, centers O r3, O r4, and representative circles of the rational numbers r, r 2, r 3, and r 4 as described in Theorem (iii).

5 Parabolas and Ford circles 79 3 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. Then we consider the finitely many Ford circles with centers on or above the horizontal line y = /2 2. Let F be the ordered set of Farey fractions of order and let N = N() be its cardinality. One knows that N() = 3 π O( log ). We now define a real valued function F on the interval [0, ] as follows. Let a/q and a /q be consecutive fractions in F. From basic properties of Farey fractions we know that the fraction θ = (a + a )/(q + q ) does not belong to F, and it is the first fraction that would be inserted between a/q and a /q if one would allow to increase. By Ford s theory, the circle C θ is tangent to both C a/q and C a /q. By its definition, C θ is also tangent to the x-axis at the point T θ = (θ, 0). We define F on the interval [a/q, a /q ] in such a way that its graph over this interval coincides with the arc of the parabola P θ lying above the same interval [a/q, a /q ]. We know from Theorem (i) that the end points of this arc are exactly the centers of the Ford circles C a/q and C a /q. This shows that the function F is well defined and continuous on the entire interval [0, ]. On each subinterval of the form [a/q, a /q ], F is given explicitly by F (t) = (q + q ) 2 (t a + ) 2 a 2 q + q. (3.) In Figure 4, we show the Ford circles and the parabolas described above in the case = 5. Thus in this case the graph of F is a union of parabolas joining those centers of Ford circles that lie on or above the line y = /(2 5 2 ) = /50. Our next goal is to estimate the area under these parabolas, that is, Area() = 0 F (t) dt. For example, in the particular case shown in Figure 4, the area under the parabolas equals Area(5) = In Table one sees how Area() changes as increases. Table : The size of Area() and Area() for selected values of.

6 80 S. Corinne Hutchinson and Alexandru Zaharescu Figure 4: The set of Ford circles and parabolas with = N () Area() Area() Since the areas appear to decay like /, in the last column we also included the quantity Area(). Based on the data from the table it is natural to conjecture that the sequence Area() converges to a certain constant, The following theorem shows that this is indeed true. Theorem 2. For all positive integers, ζ(2) Area() = +O 3ζ(3) log 2, where ζ(s) denotes the Riemann zeta function. In particular, Theorem 2 confirms the above conjecture, as lim Area() = ζ(2)/3ζ(3) =

7 Parabolas and Ford circles 8 It would be interesting to further study the oscillatory behavior of the function Area(), shown in Figure 5. Figure 5: The behavior of the function Area(). 4 Proof of theorem 2 Recall that between two consecutive Farey fractions α = a/q and β = a /q, F (t) has the form F (t) = A(t θ) 2, where θ = (a + a )/(q + q ) is the point of tangency to the x-axis and A = (q + q ) 2 /2. Thus β α F (t) dt = To simplify the integral, notice that A(t θ)3 3 β θ = a q a + a q + q = q (q + q ), α θ = a q a + a q + q = q(q + q ), β α. (4.) since by a fundamental property of Farey fractions, a q aq =. We find that the integral in (4.) reduces to β α F (t) dt = ( ) 6 q 3 (q + q ) + q 3 (q + q. ) Taking into account the symmetry of the Farey sequence with respect to /2, or in other words taking into account that for each pair of consecutive Farey fractions a/q and a /q there is another pair having the same denominators q, q, in reverse order, namely a /q and a/q, one sees that Area() = 3 q 3 (q + q ), (4.2)

8 82 S. Corinne Hutchinson and Alexandru Zaharescu where the sum is over all pairs (q, q ) of consecutive denominators of the sequence of Farey fractions of order. From basic properties of Farey fractions we know that this set of pairs of consecutive denominators coincides with the set of pairs (q, q ) of integer numbers for which q, q, gcd(q, q ) = and q + q >. Thus we may rewrite (4.2) as Area() = 3 q,q gcd(q,q )= q+q > q 3 (q + q ) := S(). (4.3) 3 For convenience, we introduce the function f(x, y) = x 3 (x+y) and the triangular region Ω = Ω(), which is defined by the inequalities x, y and x + y >. Then the sum in (4.3) can be written as S() = (x,y) Ω gcd(x,y)= f(x, y). By Möbius summation, S() = (x,y) Ω = d f(x, y) µ(d) d gcd(x,y) (x,y) Ω d x, d y µ(d) f(x, y). (4.4) We continue the evaluation in (4.4) by changing the variables x = dm and y = dn, to obtain S() = µ(d) f(dm, dn) d (m,n) d Ω = µ(d) d 4 m 3 (m + n) := (4.5) µ(d) d 4 S (d, ). d (m,n) d Ω d The inner sum S (d, ) can be written as S (d, ) = m /d = m /d m 3 m 3 /d m<n /d /d<r /d+m m + n Here we introduce a variable s to take into account the size of r near /d. We r.

9 Parabolas and Ford circles 83 obtain S (d, ) = m /d = = d = d m /d m 3 m 3 m /d m /d m s= m 3 [ d ] + s m { } s= d d + s m ( ( )) sd + O s= ( d 2 log /d m 2 + O 2 ). The completion of the sum over all positive integers m, yields S (d, ) = d m 2 d ( d 2 ) m 2 + O log /d 2 m= m>/d = dζ(2) ( d 2 ) + O log /d 2. Inserting this estimate into (4.5), we find that S() = ( ( µ(d) dζ(2) d 2 )) d 4 + O log /d 2 d = ζ(2) ζ(3) + O Hence, in view of (4.3), the proof of the theorem is completed. ( log 2 ). (4.6) 5 Application to Kunik s function In a recent paper, Kunik [8] defines a function t Ψ (t), for each positive integer, as follows. Between any two consecutive Farey fractions in F, say aj q j and a j q j, j =,..., N(), the function Ψ (t) is defined by: Ψ (t) := q j + q j 2 [ t a j + a j q j + q j ], for a j q j t < a j q j. The function Ψ (t) is a step function, being linear between subsequent Farey fractions in F and having a jump of height /q j at each Farey fraction aj q j. Kunik established a number of interesting results, including the following L 2 -estimate: ( ) log Ψ 2 2 = Ψ (t) 2 dt = O. 0

10 84 S. Corinne Hutchinson and Alexandru Zaharescu We notice a connection between the above function F (t) and Kunik s function Ψ (t). More precisely, we restate an improved L2 -estimate of Kunik s function in the following theorem. Theorem 3. For positive integers, 0 Ψ (t) 2 dt = 2 Area() = ζ(2) 6ζ(3) + O where ζ(s) denotes the Riemann zeta function. ( log The proof of this theorem is as follows. We first notice the connection between F (t) and the function Ψ (t), which says that 2 F (t) = 2Ψ (t) 2, for all t [0, ] and. From here, it is straightforward to see that by Theorem 2, one may replace the original L 2 -estimate by the one given in the theorem. Thus the proof of Theorem 3 is complete. ), References [] J. Athreya, C. Cobeli, A. Zaharescu, Radial Density in Apollonian Packings, Int. Math. Res. Notices (204) doi:0.093/imrn/rnu [2] F. Boca, C. Cobeli, A. Zaharescu, A conjecture of R. R. Hall on Farey points, J. Reine Angew. Math no. 535 (200), [3] F. Boca, A. Zaharescu, The correlations of Farey fractions, J. Lond. Math Soc. no. 72 (2005), [4] S. Chaubey, A. Malik, A. Zaharescu, k-moments of distances between centers of Ford circles, J. Math. Anal. Appl. no. 2 (205), [5] C. Cobeli, A. Zaharescu, The Haros-Farey sequence at two hundred years, Acta Univ. Apulensis Math. Inform. no. 5 (2003), [6] C. Cobeli, M. Vâjâitu, A. Zaharescu, On the intervals of a third between Farey fractions, Bull. Math. Soc. Sci. Math. Roumanie no. 3 (200), [7] L. R. Ford, Fractions, Amer. Math. Monthly no. 45 (938),

11 Parabolas and Ford circles 85 [8] M. Kunik, A scaling property of Farey fractions, Fakultät für Mathematik, Otto-Von-Guericke-Universität Magdeburg, Preprint Nr. 7 (204), Received: Revised: Accepted: Department of Mathematics, University of Illinois, 409 West Green Street, Urbana, IL 680, USA shutchi2@illinois.edu Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box -764, RO Bucharest, Romania and University of Illinois, Department of Mathematics, 409 West Green Street, Urbana, IL 680, USA zaharesc@illinois.edu