Symmetry in the sequence of approximation coefficients

Let $\{a_n\}_1^\infty$ and $\{\theta_n\}_0^\infty$ be the sequences of partial quotients and approximation coefficients for the continued fraction expansion of an irrational number. We will provide a function $f$ such that $a_{n+1} = f(\theta_{n\pm1},\theta_n)$. In tandem with a formula due to Dajani and Kraaikamp, we will write $\theta_{n \pm 1}$ as a function of $(\theta_{n \mp 1}, \theta_n)$, revealing an elegant symmetry in this classical sequence and allowing for its recovery from a pair of consecutive terms.


Introduction
Given an irrational number r and a rational number written as the unique quotient p q of the two integers p and q with gcd(p, q) = 1 and q > 0, our fundamental object of interest from diophantine approximation is the approximation coefficient θ (r, p q ) := q 2 r − p q . Small approximation coefficients suggest high quality approximations, combining accuracy with simplicity. For instance, the error in approximating π using the fraction 355 113 = 3.14159203539823008849557522124 is smaller than the error of its decimal expansion to the fifth digit 3.14159 = 314159 100000 . Since the former rational also has a much smaller denominator, it is of far greater quality than the latter. Indeed θ π, 355 113 < 0.0341 whereas θ π, 314159 100000 > 26535.
We obtain the high quality approximations for r by using the euclidean algorithm to write r as an infinite continued fraction: r = a 0 + [a 1 , a 2 , ...] := a 0 + 1 where the partial quotients a 0 = a 0 (r) ∈ Z and a n = a n (r) ∈ N := Z ∩ [1, ∞) for all n ≥ 1, are uniquely determined by r. This expansion also provides us with the infinite sequence of rational numbers p 0 q 0 := a 0 1 , p n q n := a 0 + [a 1 , ..., a n ], n ≥ 1, tending to r known as the convergents of r. Define the approximation coefficient of the n th convergent of r by θ n := θ r, p n q n = q 2 n r − p n q n and refer to the sequence {θ n } ∞ 0 as the sequence of approximation coefficients. Since adding an integer to a fraction does not change its denominator, the number x 0 := r − a 0 shares the same sequences {a n } ∞ 1 and {θ n } ∞ 0 as r, allowing us to restrict our attention solely to the unit interval. Throughout this paper, we fix an initial seed x 0 ∈ (0, 1) − Q and let {a n } ∞ 1 and {θ n } ∞ 0 be its sequences of partial quotients and approximation coefficients. While the rest of this section is not a prerequisite, the following results illustrate some of the key properties for this classical sequence and are given for motivation as well as for sake of completeness.
For all n ≥ 0, it is well known [2,Theorem 4.6] that x 0 − p n q n < 1 q n q n+1 < 1 q 2 n . We conclude that θ n < 1 for all n ≥ 0. Conversely, Legendre [2, Theorem 5.12] proved that if θ (x 0 , p q ) < 1 2 then p q is a convergent of x 0 . In 1891, Hurwitz proved that there exist infinitely many pairs of integers p and q, such that θ (x 0 , p q ) < 1 √ 5 ≈ 0.4472 and that this constant, known as the Hurwitz Constant, is sharp. Therefore, all irrational numbers possess infinitely many high quality approximations using rational numbers, whose associated approximation coefficients are less than 1 √ 5 . Using Legendre's result, we see that all these high quality approximations must belong to the sequence of continued fraction convergents for x 0 .
We may restate Hurwitz's theorem as the sharp inequality lim inf is called the Lagrange Spectrum and those irrational numbers x 0 which construe the spectrum, that is, for which lim inf n→∞ θ n (x 0 ) > 0 are called badly approximable numbers. It is known [2,Theorem 7.3] that x 0 is badly approximable if and only its sequence of partial quotients {a n } ∞ 1 is bounded. For more details about the Lagrange Spectrum, refer to [3].

Preliminary results
In 1921, Perron [6] proved that , a n+2 , ...] + a n + [a n−1 , a n−2 , ..., a 1 ], n ≥ 1, where we take [ / 0] := 0 when n = 1. Thus, as far as the flow of information goes, the entire sequence of partial quotients is needed in order to generate a single member in the sequence of approximation coefficients. In 1978, Jurkat and Peyerimhoff [5] showed that for all irrational numbers and for all n ≥ 1, the point (θ n−1 , θ n ) lies in the interior of the triangle with vertices (0, 0), (0, 1) and (1, 0). As a result, we have which is an improvement of Vahlen's result (1). In addition, they proved that a n+1 can be written as a function of (θ n−1 , θ n ) but came short of providing a simple expression, which applies to all cases. Combining this observation with the pair of symmetric identities θ n+1 = θ n−1 + a n+1 1 − 4θ n−1 θ n − a 2 n+1 θ n , n ≥ 1 and θ n−1 = θ n+1 + a n+1 1 − 4θ n+1 θ n − a 2 n+1 θ n , n ≥ 1, due to Dajani and Kraaikamp [4,proposition 5.3.6], allows us to recover the tail of the sequence of approximation coefficients from a pair of consecutive terms.
Our goal, obtained in Theorem 3, is to provide a real valued function f such that a n+1 = f (θ n±1 , θ n ).
This will enable us, as expressed in Corollary 4, to eliminate a n+1 from formula (4) without disrupting its elegant symmetry. This will enable us to recover the entire sequence {θ n } ∞ 0 from a pair of consecutive terms.

Symbolic dynamics
The continued fraction expansion is a symbolic representation of irrational numbers in the unit interval as an infinite sequence of positive integers. Let ⌊·⌋ be the floor function, whose value on a real number r is the largest integer smaller than or equal to r. Then we obtain this expansion for the initial seed x 0 ∈ (0, 1) − Q by using the following infinite iteration process: 1. Let n := 1.
2. Set the reminder of x 0 at time n to be r n := 1 x n−1 ∈ (1, ∞).
3. Define the digit and future of x 0 at time n to be the integer part and fractional part of r n respectively, that is, a n := ⌊r n ⌋ ∈ N and x n := r n − a n ∈ (0, 1) − Q. Increase n by one and go to step 2.
Using this iteration scheme, we obtain hence, the quantity a n is no other than the n th partial quotient of x 0 . We relabel a n as the digit for x 0 at time n in order to emphasis the underlying dynamical structure at hand and write The quantity x n = r n − a n is the value of x n−1 under the Gauss Map This map is realized as a left shift operator on the set of infinite sequences of digits, i.e.
We preserve the n digits that the map T n erases from this symbolic representation of x 0 by defining the past of x 0 at time n ≥ 1 to be y n := −a n − [a n−1 , a n−2 , ..., a 1 ] < −1.
The natural extension map is well defined whenever x is an irrational number and y < −1, providing us with the relationship (x n+1 , y n+1 ) = T (x n , y n ), n ≥ 1.
Since x n is uniquely determined by {a n } ∞ n+1 and y n is uniquely determined by {a n } n 1 , this map can be thought of as one tick of the clock in the symbolic representation of x 0 using the sequence {a n } ∞

Lemma 2. The map Ψ
: Ω → Γ is a homeomorphism with inverse: Proof. First, we will show that Ψ is a bijection. Since the map Ψ is surjective onto its image Γ, we need only show injectiveness. Let (x 1 , y 1 ), (x 2 , y 2 ) be two points in Ω such that .
By equating the first and then the second components of the exterior terms, we obtain that and then, that x 1 y 1 = x 2 y 2 . Therefore, Since both these points are in Ω they must lie below the line x + y = 0, hence x 1 + y 1 = x 2 + y 2 < 0. Another application of condition (14) now proves that x 1 = x 2 and y 1 = y 2 , hence Ψ is injective.
Since both Ψ and Ψ −1 are clearly continuous, it is left to prove that Ψ −1 is well-defined and that it is the inverse for Ψ.

Result
Theorem 3. Let x 0 be an irrational number in the unit interval and let n ∈ N. If a n+1 is the digit at time n + 1 in the continued fraction expansion for x 0 and if (θ n−1 , θ n , θ n+1 ) are the approximation coefficients for x 0 at time n − 1, n and n + 1, then Proof. Let (x n , y n ) be the dynamic pair of x 0 at time n. Formula (10), the fact that Ψ is a homeomorphism and the definition (13) of Ψ −1 yield Using formula (5), we write x n = [a n+1 , r n+2 ] = 1 a n+1 +[r n+2 ] , so that the first components in the exterior terms of formula (16) equate to But since [r n+2 ] = x n+1 < 1, we have a n+1 = a n+1 + [r n+2 ] = 1 + √ 1 − 4θ n−1 θ n 2θ n , which is the first equality in the equations (15).
Adding one to all indices establishes the equality of the exterior terms in the equations (15) and completes the proof.
As a direct consequence of this theorem and formula (4), we obtain: