Geodesics on modular surfaces and continued fractions

A purpose of this paper is to give a short sketch of a relation between continued fractions and the hyperbolic geometry on the upper half plane, (the simple continued fractions case and a generalized case). Relations between continued fractions and the geodesic flows on the modular surface are well-known. For example, Adler and Flatto [1] showed that the continued fraction transformation is obtained as a cross-section map of the geodesic flow. Another interesting one is due to Moeckel [8], who proved a metrical property of continued fractions concerning to a distribution of digits by using the Farey tessellation and the ergodicity of geodesic flows.

It is well-known that a continued fraction may be regarded as a sequence of Mobius maps. In this partly expository paper we consider continued fractions by examining the action of Mobius maps in hyperbolic space, and in all dimensions. We obtain a general version of the Stern-Stolz theorem valid in all dimensions, and we draw comparisons between the theory of continued fractions and the theories of discrete groups of Mobius maps, and complex dynamics.

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.

The subject of this paper is the current state of art in theory of continued fractions, intermediate fractions and their relation to the best rational approximations of the first and second kind. The paper provides an overview of the some well known and even some new properties of continued fractions, and the various terms associated with them. In addition to intermediate fractions, paper considers the fine intermediate fractions and gave some statements to position these fractions in the continued fraction representation of numbers.

The ordinary and matrix continued fractions in the theoretical analysis of Hermitian and relaxation operators

The quotient-difference (=QD) algorithm developed by the author may be considered as an extension ofBernoulli's method for solving algebraic equations. WhereasBernoulli's method gives the dominant root as the limit of a sequence of quotientsq 1 () =s 1 (+1) /s 1 () formed from a certain numerical sequences 1 () , the QD-algorithm gives (under certain conditions) all the rootsλ σ as the limits of similiar quotient sequencesq σ () =s σ (+1) /s σ () . Close relationship exists between this method and the theory of continued fractions. In fact the QD-algorithm permits developing a function given in the form of a power series into a continued fraction in a remarkably simple manner. In this paper only the theoretical aspects of the method are discussed. Practical applications will be discussed later.

Let [a 1(x), a 2(x), …, a n (x), …] be the continued fraction expansion of x ∈ [0, 1) and q n (x) be the denominator of the nth convergent. The study of relative growth rate of the product of partial quotients a n+1(x)a n (x) compared with q n (x) originated from the improvability of Dirichlet’s theorem. In this note, we prove that, for any 0 ⩽ α ⩽ β ⩽ +∞, the Hausdorff dimension of the following set Fα,β=x∈[0,1):lim infn→∞log(an+1(x)an(x))logqn(x)=α,lim supn→∞log(an+1(x)an(x))logqn(x)=β is 2β+2+β2+4 or 2β+2 according to α > 0 or α = 0, respectively. This result extends an earlier result of Huang and Wu as well as gives insights on the metric theory of Dirichlet non-improvable sets.

We revisit Ito’s [Osaka J. Math. 26 (1989), pp. 557–578] natural extension of the Farey tent map, which generates all regular continued fraction convergents and mediants of a given irrational. With a slight shift in perspective on the order in which these convergents and mediants arise, this natural extension is shown to provide an elegant and powerful tool in the metric theory of continued fractions. A wealth of old and new results—including limiting distributions of approximation coefficients, analogues of a theorem of Legendre and their refinements, and a generalisation of Lévy’s Theorem to subsequences of convergents and mediants—are presented as corollaries within this unifying theory.

On a theorem from the metric theory of continued fractions

By using a variational method, we study the structure of the Padé table for a formal power series. For series of Stieltjes, this method is employed to study the relations of the Padé approximants with orthogonal polynomials and gaussian quadrature formulas. Hence, we can study convergence, precise locations of poles and zeros, monotonicity, and so on, of these approximants. Our methods have nothing to do with determinant theory and the theory of continued fractions which were used extensively in the past.

After the first attempts of Bombelli and Cataldi, the theory of continued fractions was ready to be developed. The beginning of the theory was due mostly to Wallis, as we shall see in the next section. Another important contribution was by Huygens, who will be discussed with some other contributors.

We study the spectrum of a parameter-dependent Sturm-Liouville problem. By using, as the main tool, the theory of continued fractions we obtain a characterization of the eigenvalues. From here estimates for large eigenvalues, depending on the parameter, and an asymptotic result for the lowest eigenvalue will follow. We associate to each given eigenvalue two sequences converging monotonically to the eigenvalue itself, one from above and the other from below. To obtain this result we use the theory of orthogonal polynomials.

On the metric theory of continued fractions

and the above results also follow from the classical theory of continued fractions (for example, see [10] and the remarks in [5]). In view of this, it seems natural to attempt a study of the properties of the polynomials, Pn(x), as determined by the coefficients, cn and Xn. One approach to such an investigation is of course through the study of the convergence of the continued fraction (1.3). In this paper, however, we shall avoid direct reference to the theory of continued fractions although we shall make fairly extensive use of the results from the problem of moments. Instead, our investigation centers upon the true interval of orthogonality of Pn(x) }. [By the true interval of orthogonality, we mean, following Shohat (see [5; 6, p. 113]), the smallest interval which contains in its interior all zeros of all Pn(x).] In this connection, the chain sequences of Wall [10] arise naturally and play a fundamental role. Thus our study is largely arithmetical in character. Throughout this paper, orthogonal means polynomials

Recently, generalizations of the classical Three Gap Theorem to higher dimensions attracted a lot of attention. In particular, upper bounds for the number of nearest neighbor distances have been established for the Euclidean and the maximum metric. It was proved that a generic multi-dimensional Kronecker attains the maximal possible number of different gap lengths for every sub-exponential subsequence. We mirror this result in dimension d ∈ {2, 3} by constructing Kronecker sequences which have a surprisingly low number of different nearest neighbor distances for infinitely N ∈ ℕ. Our proof relies on simple arguments from the theory of continued fractions.