Odd-odd continued fraction algorithm

By using a jump transformation associated to the Romik map, we define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. Among others, it is proved that all the best approximating rationals of odd denominators and odd numerators of an irrational number are given by the principal convergents of the odd-odd continued fraction algorithm and vice versa.


Introduction
One main topic of Diophantine approximation studies the approximation of an irrational number by rational numbers.Given an irrational number x, we call a rational p/q a best approximation of x if |qx − p| < |bx − a| for any a b = p q such that 0 < b ≤ q.
Here, and in the whole paper, by convention, when we write a rational number p/q, we always assume that p ∈ Z, q ∈ N and p and q are coprime.The celebrated Lagrange Theorem (see [17, Chapter II] and [12,Section 6]) states that the best approximations of an irrational number x are the convergents, i.e., the finite truncations, of the regular continued fraction (RCF) of x: (1.1) x = d 0 + 1 , where d 0 ∈ Z and d j ∈ N, for j ≥ 1.More precisely, a rational p/q is a best approximation of an irrational x if and only if it is one of the convergents: Let H = {z ∈ C : Im(z) > 0} be the upper half-plane.The group SL 2 (R) acts on H as isometries defined by There is a close connection between the geodesics on the modular surface SL 2 (Z)\H and the RCF algorithm (e.g.[22]).Especially, the orbit SL 2 (Z)(∞) = Q corresponds to a unique cusp of SL 2 (Z)\H.Let Then Θ is a subgroup of SL 2 (Z) of index 3 and the quotient space Θ\H is a hyperbolic surface with two cusps corresponding to the orbits Θ(∞) and Θ(1) of ∞ and 1. Kraaikamp-Lopes [14] and Boca-Merriman [4] found that the geodesics on Θ\H is strongly related to the even integer continued fraction (EICF) introduced by Schweiger [19,20], which is a continued fraction with even integers such that , where b 0 ∈ 2Z, b i ∈ 2N for i ≥ 1 and η i ∈ {−1, +1}.
We classify rational numbers into two classes by the orbits Θ(∞) and Θ(1).If p/q ∈ Θ(∞), then p and q are of different parity.If p/q ∈ Θ(1), then p and q are both odd.We call a rational number in Θ(∞) an ∞-rational and a rational number in Θ(1) a 1-rational.The proportion of odd/even, even/odd and odd/odd in the RCF convergents was investigated by Moeckel [15].Further, the asymptotic density of the RCF convergents whose denominators and numerators satisfying congruence equations was obtained by Jager-Liadet [10].
Short and Walker [23] defined a best ∞-rational approximation of x by a rational p/q ∈ Θ(∞) satisfying and showed that the best ∞-rational approximations are convergents of EICF.
Our motivation of the paper is to study the best 1-rational approximations of an irrational number defined as follows.
We introduce a new continued fraction, called the odd-odd continued fraction (OOCF, see Section 2) of the form , where a n ∈ N, and ε n ∈ {1, −1} for a n ≥ 2 and ε n = 1 for a n = 1.Our first main theorem is the following.
Theorem 1.2.A fraction p/q is a best 1-rational approximation of an irrational number x if and only if it is one of the principal convergents of the odd-odd continued fraction of x.
For RCF, Lagrange and Euler proved that an irrational number has eventually periodic RCF if and only if it is a quadratic irrational.(See [17, Chapter III- §1] and [12,Section 10]).For OOCF, we have the following second main theorem.
Theorem 1.3.An eventually periodic OOCF expansion converges to an ∞-rational or a quadratic irrational.Moreover, a quadratic irrational has an eventually periodic OOCF expansion.
We also investigate the relation between the OOCF and the RCF.We show that for any real number x, the principal convergents of its OOCF are intermediate convergents of its RCF (Theorem 5.2).Further, we can convert RCF expansions into OOCF expansions (Theorem 5.3).
Our paper is organized as follows.In Section 2, we introduce the OOCF algorithm and give some basic properties of OOCFs.In Section 3 we study the principal convergents of OOCFs, and prove Theorem 1.3.Section 4 is devoted to the proof of Theorem 1.2.The relations between the OOCF expansions and the RCF expansions are described in the last section.

OOCF algorithm
It is known that the partial quotients d j = d j (x) of the RCF of an irrational number x as in (1.1) can be generated by the Gauss map G : [0, 1] → [0, 1] defined by where {•} is the fractional part.In fact, for an irrational x we have being the integer part.Further, Gauss map is a jump transformation associated to the Farey map defined by In general, let U : [0, 1] → [0, 1] be a map and E be a subset of [0, 1].The first hitting time of x ∈ [0, 1] to E is defined by A map J : [0, 1] → [0, 1] is called the jump transformation associated to U with respect to E (e.g.[21,Chapter 19]) if We can easily check that G is the jump transformation associated to F with respect to E = {0} ∪ (1/2, 1].
In fact, We also note that n E (x) + 1 is exactly the first partial quotient a 1 of the RCF expansion of x.
Similar to the RCF, the partial quotients of EICF in (1.2) can be obtained by the EICF map for all k ∈ N, and T EICF (0) = 0.
The map T EICF turns out to be a jump transformation of the following Romik map introduced by Romik in [18].In fact, letting E 1 := {0} ∪ [1/3, 1], we have The Romik map was used to investigate an algorithm generating the Pythagorean triples by multiplying matrices [1-3, 7, 8].Some number theoretical properties of the Romik map were recently shown in [5,6].
Panti [16] studied the connection of the Romik map with the billiards in the hyperbolic plane.
Thus, 0 has a unique infinite OOCF expansion: Then T (x) = 0 and we have both choices in (2.4): .
Note that for a rational m/n, the denominator of T OOCF (m/n) is strictly less than n.Note also that T OOCF sends a 1-rational to a 1-rational and an ∞-rational to an ∞-rational.Thus if m/n is a 1-rational, Hence any 1-rational and ∞-rational belongs to T −N OOCF ({1}) and T −N OOCF ({0}) for some N ≥ 1 respectively.Finally, we note that Then for any x ∈ [0, 1] \ Q, we can iterated (2.5) infinitely and uniquely to get its OOCF expansion: We denote the OOCF expansion of x by and call (a n , ε n ) the n-th partial quotients of x in its OOCF expansion.By the above discussion, we have the following proposition.
(2) Each 1-rational has exactly two finite OOCF expansions which differ only in the last partial quotient.
(3) Every irrational has a unique infinite OOCF expansion.
To end this section, we remark that the two maps T OOCF and T EICF are conjugate.Define f : Schweiger [19] proved that T EICF admits an ergodic absolutely continuous invariant measure: dµ := dx 1 − x 2 .Thus, the measure f −1 * µ is an ergodic absolutely continuous invariant measure with respect to T OOCF .Denote by y = f (x).We have Hence, we have the following conclusion.We also remark that the infinity of the absolutely continuous invariant measure comes from the fact that the map T OOCF has 0 as an indifferent fixed point.

Convergents of the odd-odd continued fraction algorithm
For OOCF, we have three types of convergents by truncating the OOCF in three different places.We will investigate the basic properties of such convergents of OOCF.
Let x ∈ (0, 1) be a real number such that For n ≥ 1, the n-th principal convergent of OOCF is defined by We denote and call them the n-th sub-convergent and n-th pseudo-convergent, respectively.
To study the convergents of a continued fraction, we have the following general lemma proved by induction (see [13, p. 3] for details).
Lemma 3.1.Consider a general infinite continued fraction and its truncated continued fraction of the form r n e n r n−1 s n e n s n−1 = g 0 e 0 1 0 Consequently, we have the following recursive formulas: where r −1 = 1, s −1 = 0, r 0 = g 0 and s 0 = 1.
Denote the inverse of T OOCF | B(an,εn) by f (an,εn) .Then by (2.4), we have The map f (an,εn) corresponds to a linear fractional map on the upper half-plan H given by the matrix By Lemma 3.1, we have These mean that under the linear fractional map of the matrix A (a1,ε1) A (a2,ε2) • • • A (an,εn) , the images of 1, ∞ and 0 are p n /q n , p ′ n /q ′ n and p ′′ n /q ′′ n , respectively.
Proposition 3.2.We have We remark that the name of odd-odd continued fraction comes from the fact that the principal convergents p n /q n are 1-rationals, i.e., of odd denominators and odd numerators.We also remark that by Proposition 3.2, any finite OOCF is a 1-rational.By (3.3) and (3.4), we have the following recursive relations of the three types of convergents.Lemma 3.3.Let p ′ 0 = 1, q ′ 0 = 0, p 0 = 1 and q 0 = 1.We have the following recursive formulas: Further, Moreover, letting p −1 = −1, q −1 = 1 and ε 0 = 1, we have the recursive formulas for the principal convergents for n ≥ 1.
By (3.6), we have for all n ∈ N. Thus, The following theorem gives the convergence of our OOCF.
Proof.The equations (3.3) and (3.4) imply that p , which means that x − p n /q n and x − p ′′ n /q ′′ n has opposite signs by comparing with (3.12).
With the above preparations, we can now show the following lemma.
(1) The n-th principal convergent p n /q n is between p ′ n /q ′ n and p ′′ n /q ′′ n .
Proof.The first two assertions follow from (3.6) and the fact that for two rationals a/b and c/d For (3), by Lemma 3.3, we have and Since a n − 1 ≥ 0 and a n + ε n − 1 ≥ 0, both p ′ n /q ′ n and p ′′ n /q ′′ n are in I n−1 .By the first assertion, p n /q n is also in I n−1 .Now, we are ready to prove Theorem 1.3.
Proof of Theorem 1.3.If x has an eventually periodic OOCF, then there exist distinct positive integers i and j such that T i (x) = T j (x).Since T i and T j are linear fractional maps, T i (x) is either 0 or a quadratic irrational.In the former case x is an ∞-rational, while in the latter case x is a quadratic irrational.
For the second assertion, let x be a quadratic irrational between 0 and 1 such that α 1 x 2 + β 1 x + γ 1 = 0 where α 1 , β 1 and γ 1 are coprime integers.By (3.10), we have for all i ≥ 1, (3.13) Then On the other hand, we also have Thus, by Lemma 3.5 and the fact that p i /q i is between p ′ i /q ′ i and p ′′ i /q ′′ i , we deduce that x − p ′ i /q ′ i and x − p ′′ i /q ′′ i have opposite signs.Thus, Hence, there are |δ| < 1 and |λ| < 1 such that (3.16) By plugging (3.16) in (3.13), we derive the following expressions (3.17) , α i+1 is also bounded.Thus in all cases, the coefficients of the equation (3.14) are all bounded.Therefore, {ζ i } i∈N has only finitely many values which means that ζ n = ζ m for some m and n.Therefore, the OOCF expansion of x is eventually periodic.Remark 3.7.From the proof of Proposition 2.1-(1), we see that if x is an ∞-rational, then its OOCF ends with (2, −1) ∞ and thus there exists n 0 ≥ 0 such that ζ n+1 = 0 for all n ≥ n 0 .Hence, by (3.10), we have x = p ′′ n /q ′′ n for all n ≥ n 0 .
At the end of this section, let us discuss the relation between the OOCF convergents of a number x and the EICF convergents of 1 − x.
We denote the EICF expansion in (1.2) by a sequence in a double angle bracket: The n-th EICF convergent is denoted by By Lemma 3.1, we have the following matrix relation: Proposition 3.8.Let x ∈ (0, 1).All rationals of type odd/odd in {1 − p E n (1 − x)/q E n (1 − x) : n ≥ 1} are best 1-rational approximations of x, and hence are OOCF principal convergents of x.
Proof.For each n ≥ 0, denote by We have By the theorem of Short and Walker (see (1.3); also [23,Theorem 5]), for any a/b ∈ Θ(∞) such that The next proposition describes a connection between the principal convergents of OOCF and EICF.Recall that f (x) = 1−x 1+x is the conjugacy map defined in Section 2.
Considering the finite expansions, we obtain the second assertion.

Best 1-rational approximation
In this section, we prove Theorem 1.2.Recall that H is the upper half-plane.The boundary of H is With these preparations, we are ready to prove our Theorem 1.2.Proof of Theorem 1.2.Given x ∈ R \ Q, let us consider its n-th principal convergent p n /q n and its n-th pseudo-convergent p ′′ n /q ′′ n .Let a/b ∈ Θ(1) such that a/b = p n /q n and 1 ≤ b ≤ q n .Then n are tangent to each other.By Lemma 3.5, x is between p n /q n and p ′′ n /q ′′ n , then (4.4) R pn/qn (x) ≤ rad(C p ′′ n /q ′′ n ).Let I n be the closed interval of endpoints p n /q n and p ′′ n /q ′′ n .Since rad(C r/s ) ≤ rad(C pn/qn ) for any r/s ∈ I n ∩ Q, we deduce from (4.3) (as shown in Figure 3) that a/b ∈ I n .Then we have (4.5)rad(C p ′′ n /q ′′ n ) < R a/b (x).By (4.4) and (4.5), we have R pn/qn (x) < R a/b (x).Hence by (4.1), p n /q n is a best 1-rational approximation of x.Conversely, assume that a/b ∈ Θ(1) is not a principal convergent of OOCF of x.Then there are consecutive principal convergents p n−1 /q n−1 and p n /q n such that q n−1 ≤ b < q n and a/b = p n−1 /q n−1 .Thus, n .Without loss of generality, we assume that p n−1 /q n−1 < p ′ n /q ′ n < p ′′ n /q ′′ n (see Figure 4).By (4.6), a/b ∈ [p ′ n /q ′ n , p ′′ n /q ′′ n ].Now we will show that R pn−1/qn−1 (x) < R a/b (x) which by implies that a/b is not a best 1-rational approximation of x.We distinguish three cases.
(2) Now assume a/b > p ′′ n /q ′′ n .We note that, for r/s ∈ Q and t, t ′ ∈ R, Then we have The first and last inequalities in (4.8) follow from (4.7) and the fact p n−1 /q n−1 < x < p ′′ n /q ′′ n < a/b.The equality in (4.8) holds since C p ′′ n /q ′′ n and C pn−1/qn−1 are tangent to each other.The second last inequality in (4.8) follows from (4.2).
(3) Finally, let a/b ∈ (p n−1 /q n−1 , p ′ n /q ′ n ).Denote by C and C ′ the horocycles based at x tangent to C a/b and C pn−1/qn−1 , respectively (see Figure 4).Since the tangent point of C and C a/b is an interior point of the area bounded by C pn−1/qn−1 , C p ′ n /q ′ n and the real line, we conclude that C intersects C pn−1/qn−1 .Thus, C is larger than C ′ , i.e, R pn−1/qn−1 (x) < R a/b (x).

Relation with the regular continued fraction
For simplicity, we denote a RCF as in ( are called the intermediate convergents (see [12,Section 6] and [17, p.36]).Kraaikamp and Lopes [14] showed that the convergents of EICF are intermediate convergents of RCF.In this section, we show that the OOCF principal convergents are also intermediate convergents of RCF.
The following lemma tells us how the piecewise inverses of OOCF act on RCF expansions. .
By Lemma 5.1, x and p k /q k have the same prefix in their RCF expansions, except for the last partial quotient of p k /q k .Thus, p k /q k is an intermediate convergent of x.
Next, we show that we can convert RCF expansions into OOCF expansions.Before we state the theorem, let us introduce the following notations:

E2Figure 1 .
Figure 1.The graph of R (left) and the graph of T OOCF (right)

,
where g n ∈ Z and |e n | = 1.Then the following matrix relation holds:

2 |bx − a| 2 .
Denote by C a/b the horocycle of H based at a/b whose Euclidean radius is (2b 2 ) −1 and C ∞ the line {z = x + i ∈ H : x ∈ R} (see Figure 2).We call C a/b a Ford circle.The radius of C a/b is denoted by rad(C a/b ).We remark that two Ford circles C a/b and C c/d are adjacent to each other if and only if |ad − bc| = 1.Let R a/b (x) be the Euclidean radius of the horocycle based at x tangent to C a/b .Then by Pythagorean theorem, R a/b (x) = 1 Thus |qx − p| < |bx − a| ⇐⇒ R p/q (x) < R a/b (x).(4.1)Since the Ford circles are not overlapped each other, we have (4.2) rad(C a/b ) ≤ R c/d (a/b) for all a/b, c/d ∈ Q.

Figure 2 .Figure 3 .
Figure 2. Ford circles: white circles are based at ∞-rationals and gray circles are based at 1-rationals

Figure 4 .
Figure 4.A possible relative position of x, a/b and the convergents.The dashed circles C and C ′ are horocycles based at x tangent to C a/b and C pn−1/qn−1 .