Continued fractions with multiple limits

The connections between the Golden Ratio namely (1 + √5)/2, a simple continued fraction, and Fibonacci and Lucas numbers, are familiar. The Fibonacci and Lucas numbers have many fascinating properties. We now point out that the square root of the Golden Ratio is the real part of a simple periodic continued fraction but using (complex) Gaussian integers a + ib instead of the natural integers. This fact provokes a definition and a study of complex Fibonacci and Lucas numbers, and the study again turns out to have a rich theoretic structure. A fuller account will appear in The Fibonacci Quarterly.

In this paper we derive a relation concerning the speed of convergence of the nearest integer and regular continued fractions. If A n / B n {A_n}/{B_n} , p k / q k {p_k}/{q_k} denote the convergents of the nearest integer and regular continued fractions of an irrational number α \alpha , then for all n there is a k ( n ) k(n) such that A n / B n = p k ( n ) / q k ( n ) {A_n}/{B_n} = {p_{k(n)}}/{q_{k(n)}} . It is shown that \[ lim n → ∞ n k ( n ) = log ⁡ ( 1 + 5 2 ) log 2 \lim \limits _{n \to \infty } \;\frac {n}{{k\left ( n \right )}} = \frac {{\log \left ( {\frac {{1 + \sqrt 5 }}{2}} \right )}}{{\log \;2}} \] for almost all α \alpha . This problem is reduced to a special case of a general result concerning the frequency of partial quotients in the regular continued fraction (Theorem 2).

We study the appearance and properties of minimal residual fractions of polynomials in the decomposition of algebraic numbers into continued fractions. It is shown that for purely real algebraic irrationalities ???? of degree ???? > 2, starting from some number ????0 = ????0(????), the sequence of residual fractions ???????? is a sequence of given algebraic irrationalities. The definition of the generalized number of Piso, which differs from the definition of numbers he’s also the lack of any requirement of integrality. It is shown that for arbitrary real algebraic irrationals ???? of degree ???? > 2, starting from some number ????0 = ????0(????), the sequence of residual fractions ???????? is a sequence of generalized numbers Piso. Found an asymptotic formula for the conjugate number to the residual fractions of generalized numbers Piso. From this formula it follows that associated to the residual fraction ???????? are concentrated about fractions − ????????−2 ????????−1 is either in the interval of radius ???? (︁ 1 ????2 ????−1 )︁ in the case of purely real algebraic irrationals, or in circles with the same radius in the General case of real algebraic irrationals, which have complex conjugate of a number. It is established that, starting from some number ????0 = ????0(????), fair recurrent formula for incomplete private ???????? expansions of real algebraic irrationals ????, Express ???????? using the values of the minimal polynomial ????????−1(????) for residual fractions ????????−1 and its derivative at the point ????????−1. Found recursive formula for finding the minimal polynomials of the residual fractions using fractional-linear transformations. Composition this fractional-linear transformation is a fractional-linear transformation that takes the system conjugate to an algebraic irrationality of ???? in the system of associated to the residual fraction, with a pronounced effect of concentration about rational fraction − ????????−2 ????????−1 . It is established that the sequence of minimal polynomials for the residual fractions is a sequence of polynomials with equal discriminantly. In conclusion, the problem of the rational structure of a conjugate of the spectrum of a real algebraic irrational number ???? and its limit points.

Let q, fj, t2, be a sequence of elements of a field F. We give a 2 3 continued fraction algorithm for t^x + txx + t2x + .If our sequence satisfies a linear recurrence, then the continued fraction algorithm is finite and produces this recurrence.More generally the algorithm produces a nontrivial solution of the system s ',+/*/.0</<*-l, /=0 for every positive integer s. 1.Let t0, tx, t2, be a sequence of elements of a field F. Set T-\ttft./=0 Let d be a nonnegative integer.We say that T* is an approximation of T of degree d if there exist polynomials V and W such that T* = V/W, deg V < d, deg W < d, x\w, and x2d\WT -V.We shall give a continued fraction expansion for xT.This yields polynomials V-, W, and integers d-, 0 = dx < d2 < d3 < , such that (V, W) = 1 and V,/W-is an approximation of T of degree d.Suppose T* is any approximation of T of some degree d.Then T* = V/W for that value of i such that d < d<di+x.If the sequence of the t satisfies a linear recurrence of degree d, but not one of smaller degree, then there is an m such that dm = d and the linear recurrence is given by the polynomial Wm.In this case, WmT = Vm, the continued fraction expansion, terminates at i = m, and we can determine Wm from the first 2d of the t:, i.e., from those t such that 0 </ < 2d.Our algorithm is closely related to Zierler's version of Berlekamp's algorithm [1].2. We consider continued fraction expansions of the form a = A, +-l A, + 2 A3+-where A,, A2, A3, are elements from some field E. We can write

Let ξ be a real number and let n be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents w n (ξ) and w n * (ξ) defined by Mahler and Koksma. We calculate their six values when n=2 and ξ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction ξ by quadratic surds.

Ramanujan recorded many beautiful continued fractions in his notebooks. In this paper, we derive several indentities involving the Ramanujan continued fraction c(q), including relations between c(q) and c(qn). We also obtain explicit evaluations of for various positive integers n.

Based on work done by Morse and Hedlund (1940) it was observed by Arnoux and Rauzy (1991) that the classical continued fraction algorithm provides a surprising link between arithmetic and diophantine properties of an irrational number $\alpha$, the rotation by $\alpha$ on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and combinatorial properties of the well known Sturmian sequences, a class of sequences on two letters with low subword complexity. It has been conjectured since the early 1990ies that this correspondence carries over to generalized continued fraction algorithms, rotations on higher dimensional tori, and so-called $S$-adic sequences generated by substitutions. The idea of working towards this generalization is known as Rauzy’s program. Although, starting with Rauzy (1982) a number of examples for such a generalization was devised, Cassaigne, Ferenczi, and Zamboni (2000) came up with a counterexample that showed the limitations of such a generalization. Nevertheless, recently Berthe, Steiner, and Thuswaldner (2016) made some further progress on Rauzy’s program and were able to set up a generalization of the above correspondences. They proved that the above conjecture is true under certain natural conditions. A prominent role in this generalization is played by tilings induced by generalizations of the classical Rauzy fractal introduced by Rauzy (1982). Another idea which is related to the above results goes back to Artin (1924), who observed that the classical continued fraction algorithm and its natural extension can be viewed as a Poincare section of the geodesic flow on the space $SL_2(\mathbb{Z}) \ SL_2(\mathbb{R})$. Arnoux and Fisher (2001) revisited Artin’s idea and showed that the above mentioned correspondence between continued fractions, rotations, and Sturmian sequences can be interpreted in a very nice way in terms of an extension of this geodesic flow which they called the scenery flow. Currently, Arnoux et al. are setting up elements of a generalization of this connection as well. It is the aim of my series of lectures to review the above results.

We prove that if the continued fraction $K(a_{n}/1)$ has circular twin value sets $\langle V_{0},V_{1}\rangle$, then $K(a_{n}/1)$ converges except in some very special cases. The results generalize previous work by Jones and Thron.

In his paper "Multidimensional continued fractions" ( Ann. Univ. Sci. Budapest. Eötvös Sect. Math. , v. 13, 1970, pp. 113-140), G. Szekeres introduced a new higher dimensional analogue of the ordinary continued fraction expansion of a single real number. The Szekeres algorithm associates with each k -tuple ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) of real numbers (satisfying 0 &gt; α i &gt; 1 0 &gt; {\alpha _i} &gt; 1 ) a sequence b 1 , b 2 , … {b_1}, {b_2}, \ldots of positive integers; this sequence is called a continued k -fraction, and for k = 1 it is just the sequence of partial quotients of the ordinary continued fraction for α 1 {\alpha _1} . A simple recursive procedure applied to b 1 , b 2 , … {b_1}, {b_2}, \ldots produces a sequence a ( n ) = ( A n ( 1 ) / B n , … , A n ( k ) / B n ) ( n = 1 , 2 , … ; A n ( i ) ⩾ 0 a(n) = (A_n^{(1)}/{B_n}, \ldots ,A_n^{(k)}/{B_n})\;(n = 1,2, \ldots ;A_n^{(i)} \geqslant 0 and B n &gt; 0 {B_n} &gt; 0 are integers) of simultaneous rational approximations to ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) and a sequence c ( n ) = ( c n 0 , c n 1 , … , c n k ) ( n = 1 , 2 , … ) c(n) = ({c_{n0}},{c_{n1}}, \ldots ,{c_{nk}})\;(n = 1,2, \ldots ) of integer ( k + 1 ) (k + 1) -tuples such that the linear combination c n 0 + c n 1 α 1 + ⋯ + c n k α k {c_{n0}} + {c_{n1}}{\alpha _1} + \cdots + {c_{nk}}{\alpha _k} approximates zero. Szekeres conjectured, on the basis of extensive computations, that the sequence a ( 1 ) , a ( 2 ) , … a(1),a(2), \ldots contains all of the "best" simultaneous rational approximations to ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) and that the sequence c ( 1 ) , c ( 2 ) , … c(1),c(2), \ldots contains all of the "best" approximations to zero by the linear form x 0 + x 1 α 1 + ⋯ + x n α n {x_0} + {x_1}{\alpha _1} + \cdots + {x_n}{\alpha _n} . For the special case k = 2 and α 1 = θ 2 − 1 , α 2 = θ − 1 {\alpha _1} = {\theta ^2} - 1,{\alpha _2} = \theta - 1 (where θ = 2 cos ⁡ ( 2 π / 7 ) \theta = 2\cos (2\pi /7) is the positive root of x 3 + x 2 − 2 x − 1 = 0 {x^3} + {x^2} - 2x - 1 = 0 , Szekeres further conjectured that the 2-fraction b 1 , b 2 , … {b_1},{b_2}, \ldots is "almost periodic" in a precisely defined sense. In this paper the Szekeres conjectures concerning best approximations to zero by the linear form x 0 + x 1 ( θ 2 − 1 ) + x 2 ( θ − 1 ) {x_0} + {x_1}({\theta ^2} - 1) + {x_2}(\theta - 1) and concerning almost periodicity for the 2-fraction of ( θ 2 − 1 , θ − 1 ) ({\theta ^2} - 1,\theta - 1) are proved. The method used can be applied to other pairs of cubic irrationals α 1 , α 2 {\alpha _1},{\alpha _2} .

On the residue classes of integer sequences satisfying a linear three term recurrence formula

Throughout the present Chapter, we are essentially concerned with the following problem: for which functions Ψ : ℝ ≥1 → : ℝ ≥0 is it true that, for a given real number ξ, or for all real numbers ξ in a given class, the equation |ξ – p/q | q ) has infinitely many solutions in rational numbers p/q ? We begin by stating the results on rational approximation obtained by Dirichlet and Liouville in the middle of the nineteenth century. In Section 1.2, we define the continued fraction algorithm and recall the main properties of continued fractions expansions. These are used in Section 1.3 to give a full proof of a metric theorem of Khintchine. The next two Sections are devoted to the Duffin–Schaeffer Conjecture and to some complementary results on continued fractions. Dirichlet and Liouville Every real number ξ can be expressed in infinitely many ways as the limit of a sequence of rational numbers. Furthermore, for any positive integer b , there exists an integer a with |ξ – a/b | ≤ 1/(2 b ), and one may hope that there are infinitely many integers b for which |ξ – a/b | is in fact much smaller than 1/(2 b ). For instance, this is true when ξ is irrational, as follows from the theory of continued fractions.

For a complex polynomial $D(t)$ of even degree, one may define the continued fraction of $\sqrt{D(t)}$. This was found relevant already by Abel in 1826, and later by Chebyshev, concerning integration of (hyperelliptic) differentials; they realized that, contrary to the classical case of square roots of positive integers treated by Lagrange and Galois, we do not always have pre-periodicity of the partial quotients. In this paper we shall prove that, however, a correct analogue of Lagrange's theorem still exists in full generality: pre-periodicity of the {\it degrees} of the partial quotients always holds. Apparently, this fact was never noted before. This also yields a corresponding formula for the degrees of the convergents, for which we shall prove new bounds which are generally best possible (halving the known ones). We shall further study other aspects of the continued fraction, like the growth of the heights of partial quotients. Throughout, some striking phenomena appear, related to the geometry of (generalized) Hyperelliptic Jacobians. Another conclusion central in this paper concerns the poles of the convergents: there can be only finitely many rational ones which occur infinitely many times. (This is crucial for applications to a function field version of a question of McMullen.) Our methods rely, among other things, on linking Pade approximants and convergents with divisor relations in generalized Jacobians; this shall allow an application of a version for algebraic groups, proved in this paper, of the Skolem-Mahler-Lech theorem.

Introduction Numerous elementary and higher transcendental functions of complex analysis possess representations in terms of continued fractions of the Stieltjes type. (An abundance of examples are given in the textbooks by WALL [1948] and PERRON [t957]; for some recent examples and applications, see MAEHLY [1960], CONTE and FRIED [t96t], MILLER, MOLMUD and MEECHAM [t962], BLANCH [1964], PATRY and KELLER [1964], W¥1~N [1964].) Similar representations also exist for the remainder terms of the asymptotic expansions of functions defined by certain Laplacian integrals (HESRICI [t963, 1965]). Such representations are of considerable computational value, both because of the ease with which a continued fraction can be evaluated recursively, and because of the fact that continued fraction representations frequently are convergent (and computationally efficient) in wider regions of the complex plane than, say, a Taylor series. In actual computation it is necessary to approximate the value of a continued fraction by some approximant of finite order. This paper makes an a t tempt to s tudy the error committed in this approximation. Both a posteriori bounds depending on the knowledge of a finite number of approximants -and a priori bounds depending only on the elements of the continued fraction or on the coefficients of its corresponding power series are presented. A solution of the problem of finding a posteriori bounds which in a sense is best possible is given in section 2. Several different a priori bounds are given in section 3. Although these bounds are not shown to be best possible, they are shown to have the correct order of magnitude in some important special cases. The tack of a generally accepted terminology and even of clear definitions in the analytic theory of continued fractions made it necessary to recollect, in a unified notation, some basic facts of the theory in section t.

Suppose that some harmonic analysis arguments have been invoked to show that the indicator function of a set of residue classes modulo some integer has a large Fourier coefficient. To get information about the structure of the set of residue classes, we then need a certain type of complementary result. A solution to this problem was given by Gregory Freiman in 1961, when he proved a lemma which relates the value of an exponential sum with the distribution of summands in semi-circles of the unit circle in the complex plane. Since then, Freiman’s Lemma has been extended by several authors. Rather than residue classes, one has considered the situation for finitely many arbitrary points on the unit circle. So far, Lev is the only author who has taken into consideration that the summands may be bounded away from each other, as is the case with distinct residue classes. In this paper, we extend Lev’s result by lifting a recent result of ours to the case of the points being bounded away from each other.

converge to P(z) uniformly over an arbitrary closed region, K, exterior to the real axis. (ii) If P(z) is a Stieltjes power series with radius of convergence R>0, then an arbitrary infinite sequence of distinct approximants converges to P(z) uniformly over the circle Iz -=R-6, a > 0. (iii) If P(z) is a positive definite power series which is summable (Borel) to F(z), then Sk(k > 1) converges to F(z) uniformly over K. If the reciprocal of P(z) is summable (Borel) the same holds for k < 1. (iv) If P(z) is a Stieltjes series possessing minimal extensionsi D=-n Cif(-z)i for every n, and if the series EZiXo(1)ic-ilz-i-' is summable (Borel) to F(z), then in the Pade table for P(z), S-1 converges to F(z) uniformly over an arbitrary closed region exterior to the negative half of the real axis. As a corollary to (iii) we find that when P(z) is summable (Borel) and has a corresponding continued fraction, the latter is necessarily convergent to the Borel sum. The converse is not true. That is, the corresponding continued fraction may converge when P(z) is not summable (Borel). 2. The convergence of the diagonal files in the Pade table for a convergent positive definite power series. We recall that P(z) is a positive definite series if all the quadratic forms j= n= , 1, 2, * , are positive de-